#444623

order	512
nilpotency class	3
isoclinism	isoclinic class of #444622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15784
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,14,8,12,7,13)(3,15,19,41,20,9,18,33)(4,21,24,10,25,37,30,36)(5,26,28,39,29,40,27,11)(16,43,22,31,46,34,45,42)(17,47,48,32,50,57,49,38)(23,53,55,51,56,35,54,58)(44,62,52,61,63,59,64,60);(1,3)(2,9)(4,22)(5,17)(6,18)(7,19)(8,20)(10,34)(11,38)(12,15)(13,33)(14,41)(16,24)(21,42)(23,52)(25,45)(26,57)(27,48)(28,49)(29,50)(30,46)(31,37)(32,39)(35,61)(36,43)(40,47)(44,55)(51,59)(53,60)(54,63)(56,64)(58,62);(1,4,6,24,8,25,7,30)(2,10,13,21,12,36,14,37)(3,16,19,22,20,46,18,45)(5,23,28,55,29,56,27,54)(9,31,41,43,15,42,33,34)(11,35,40,51,39,53,26,58)(17,44,48,52,50,63,49,64)(32,59,47,61,38,62,57,60);(1,5,8,29)(2,11)(3,17,20,50)(4,23,25,56)(6,28,7,27)(9,32)(10,35)(12,39)(13,40)(14,26)(15,38)(16,44,46,63)(18,49,19,48)(21,51)(22,52,45,64)(24,55,30,54)(31,59)(33,57)(34,60)(36,53)(37,58)(41,47)(42,62)(43,61)

Polycyclic presentation:

2^9|a^2=g,c^2=eg,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264921, #72265041
maximal normal subgroup of	#72264921, #72265041
maximal quotient of	#72314055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351819);