#7747298

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

4642 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)(11,17)(15,21)(18,23)(22,24)(25,26,31,32)(27,35,29,33)(28,38,39,34)(30,36,40,37);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(26,32)(27,29)(34,38)(36,37);(10,14)(15,21)(16,19)(22,24)(25,27,31,29)(26,33,32,35)(28,36,39,37)(30,34,40,38);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,31,39)(26,34,32,38)(27,37,29,36)(30,35,40,33);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,27)(26,35,32,33)(28,36,39,37)(30,34,40,38);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,30,31,40)(26,36,32,37)(27,38,29,34)(28,35,39,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7654494);