#7969461

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

4571 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)(11,15)(12,22)(13,18)(14,16)(17,19)(20,21)(23,24)(25,26)(27,35)(28,38)(29,34)(30,32)(31,37)(33,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,24)(21,22)(25,27,30,37)(26,31,32,35)(28,36,39,29)(33,40,38,34);(10,16)(11,19)(12,18)(21,23)(26,32)(31,35)(33,38)(34,40);(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,24)(22,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(9,12)(10,13)(11,20)(14,18)(15,23)(16,22)(17,21)(19,24)(25,28)(26,33)(27,36)(29,37)(30,39)(31,40)(32,38)(34,35);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,21)(12,16)(14,22)(15,20)(17,24)(19,23)(25,29,30,36)(26,34,32,40)(27,28,37,39)(31,33,35,38)

Polycyclic presentation:

2^9|b^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876657);