#6440165

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2157 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)(13,14,17,22)(15,24,26,19)(16,21,28,37)(18,31,30,23)(20,32,36,29)(25,39,41,34)(27,35,43,40)(33,42,38,44);(1,2)(3,4)(6,9)(10,12)(13,15,17,26)(14,19,22,24)(16,25,28,41)(18,27,30,43)(20,33,36,38)(21,34,37,39)(23,35,31,40)(29,42,32,44);(2,4)(13,16,17,28)(14,20,22,36)(15,25,26,41)(18,32,30,29)(19,33,24,38)(21,23,37,31)(27,44,43,42)(34,35,39,40);(2,4)(13,16)(14,21)(15,25)(17,28)(18,29)(19,34)(20,31)(22,37)(23,36)(24,39)(26,41)(27,42)(30,32)(33,40)(35,38)(43,44);(5,7,8,11)(6,10,9,12)(16,28)(20,36)(21,37)(25,41)(29,32)(33,38)(34,39)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,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, 6347361);