#8091695

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

4765 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7998891);