#8425138

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

4593 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332334);