#8504739

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

4350 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)(4,6)(5,8)(9,10,15,13)(11,18,16,12)(14,17,23,21)(19,22,24,20)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,3,4,8)(2,5,6,7)(10,13)(12,18)(17,21)(20,22)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,22)(23,24)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,2,4,6)(3,5,8,7)(9,12,15,18)(10,16,13,11)(14,22,23,20)(17,19,21,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,3)(2,5)(4,8)(6,7)(9,13,15,10)(11,18,16,12)(14,21,23,17)(19,22,24,20);(1,2)(3,5)(4,6)(7,8)(9,14,15,23)(10,17,13,21)(11,19,16,24)(12,20,18,22)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411935);