#6456572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 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:

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

Polycyclic presentation:

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

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, 6363768);