#8093710

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

4432 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000906);