#8695299

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

4400 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

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, 8602495);