#6440100

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	#24438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2173 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)(13,14,18,24)(15,25,28,20)(16,22,31,37)(17,23,19,33)(21,32,38,30)(26,40,43,35)(27,41,29,36)(34,42,39,44);(1,2)(3,4)(6,9)(10,12)(13,15,18,28)(14,20,24,25)(16,26,31,43)(17,27,19,29)(21,34,38,39)(22,35,37,40)(23,36,33,41)(30,42,32,44);(2,4)(13,16,18,31)(14,21,24,38)(15,26,28,43)(17,30,19,32)(20,34,25,39)(22,23,37,33)(27,42,29,44)(35,36,40,41);(2,4)(13,16)(14,22)(15,26)(17,32)(18,31)(19,30)(20,35)(21,33)(23,38)(24,37)(25,40)(27,44)(28,43)(29,42)(34,41)(36,39);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,24,33)(15,27,28,29)(16,30,31,32)(20,36,25,41)(21,37,38,22)(26,42,43,44)(34,40,39,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,g^c=gi,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, 6347296);