#8696205

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=g,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, 8603401);