#8097261

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

4520 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004457);