#6467406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,19,33,18,22,17,23)(15,24,32,20,26,39,27,36)(16,28,34,38,30,21,31,43)(25,41,44,37,29,40,42,35);(2,3)(13,15)(14,20)(16,29)(17,32)(18,26)(19,27)(21,37)(22,36)(23,39)(24,33)(25,30)(28,35)(31,42)(34,44)(38,40)(41,43);(1,2,4,3)(6,9)(10,12)(13,16,18,30)(14,21,22,28)(15,25,26,29)(17,31,19,34)(20,35,36,37)(23,38,33,43)(24,40,39,41)(27,42,32,44);(1,3,4,2)(14,22)(15,26)(21,28)(23,33)(25,29)(27,32)(38,43)(42,44);(5,7,8,11)(6,10,9,12)(16,30)(21,28)(25,29)(31,34)(35,37)(38,43)(40,41)(42,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374602);