#8662975

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8570171);