#7961798

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7868994);