#8664932

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

4505 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)(11,18,21,16)(12,17,13,22)(19,24,20,23)(25,26)(27,36)(28,35)(29,34)(30,32)(31,33)(37,39)(38,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,19)(13,20)(14,18)(15,21)(17,23)(22,24)(25,27,32,31)(26,33,30,36)(28,37,29,38)(34,39,35,40);(1,3,5,6)(2,4,7,8)(9,12,15,13)(10,17,14,22)(11,19,21,20)(16,23,18,24)(25,28,32,29)(26,34,30,35)(27,37,31,38)(33,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,12)(10,17,14,22)(11,20,21,19)(16,23,18,24)(25,29,32,28)(26,35,30,34)(27,37,31,38)(33,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,18)(12,19)(13,20)(14,16)(15,21)(17,24)(22,23)(25,30,32,26)(27,36,31,33)(28,35,29,34)(37,40,38,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,10)(11,18,21,16)(12,22,13,17)(19,24,20,23)(25,31,32,27)(26,33,30,36)(28,38,29,37)(34,39,35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572128);