#8695335

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

4438 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8602531);