#7918691

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

4509 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)(11,19)(12,21)(13,18)(14,16)(15,20)(17,22)(23,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24);(1,3,5,4)(2,7,6,8)(10,16)(11,20)(17,21)(18,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,12,14,22)(10,17,16,21)(11,18,20,24)(13,19,23,15)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,22)(20,21)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,e^2=h,f^2=gh,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, 7825887);