#7961800

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

4465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=hi,f^2=gi,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, 7868996);