#8152151

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

4452 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:

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

Polycyclic presentation:

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

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, 8059347);