#6467802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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)(13,14,18,22)(15,23,24,38)(16,20,28,36)(17,30,31,21)(19,32,33,26)(25,39,27,42)(29,44,40,37)(34,43,35,41);(2,3)(13,15,17,26,18,24,31,32)(14,19,21,23,22,33,30,38)(16,27,29,43,28,25,40,41)(20,35,37,42,36,34,44,39);(1,2,4,3)(6,9)(10,12)(15,24)(16,28)(19,33)(20,36)(23,38)(26,32)(29,40)(37,44);(1,3,4,2)(13,16)(14,20)(15,25)(17,29)(18,28)(19,34)(21,37)(22,36)(23,39)(24,27)(26,41)(30,44)(31,40)(32,43)(33,35)(38,42);(5,7,8,11)(6,10,9,12)(16,28)(20,36)(25,27)(29,40)(34,35)(37,44)(39,42)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374998);