#7711854

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

4602 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)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(29,37)(32,36)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,38,40,39);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(17,20)(19,24)(26,28)(31,33)(32,36)(38,39);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619050);