#6575725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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)(15,24)(16,28)(17,22)(18,23)(19,32)(20,26)(21,29)(25,40)(27,36)(30,37)(31,35)(33,38)(34,42)(39,43)(41,44);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,20,23,24)(16,25,30,41)(17,31,19,27)(21,34,38,39)(22,36,32,35)(28,44,37,40)(29,43,33,42);(2,4)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,34)(22,37)(23,38)(24,39)(26,41)(27,42)(28,32)(31,43)(35,44)(36,40);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,26)(20,24)(25,41)(27,31)(34,39)(35,36)(40,44)(42,43);(2,4)(13,17,18,19)(14,22,23,32)(15,27,26,31)(16,29,30,33)(20,35,24,36)(21,37,38,28)(25,42,41,43)(34,44,39,40)

Polycyclic presentation:

2^9|b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482921);