#8077307

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

4675 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)(9,10,14,16)(11,17,21,15)(12,18,23,13)(19,20,22,24)(25,26,29,30)(27,31,35,33)(28,32,36,38)(34,39,37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(10,16)(13,18)(15,17)(20,24)(26,30)(28,36)(31,33)(34,37);(9,11)(10,17)(12,22)(13,20)(14,21)(15,16)(18,24)(19,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,36)(38,39);(9,12)(10,18)(11,19)(13,16)(14,23)(15,24)(17,20)(21,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,18)(10,12,16,23)(11,20,21,24)(15,19,17,22)(25,28,29,36)(26,32,30,38)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,f^2=g,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, 7984503);