#407049

order	512
nilpotency class	3
isoclinism	isoclinic class of #407048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14946
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

1115 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1115 conjugacy classes of subgroups — too many to draw. Only its 224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314245);