#8097735

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

4433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=g,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, 8004931);