#444621

order	512
nilpotency class	3
isoclinism	isoclinic class of #444620
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

1051 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1051 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306351, #72306450, #72306467
maximal normal subgroup of	#72306351, #72306450, #72306467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351817);