#6596925

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	C₂×#3271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2193 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6504121);