#8664918

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

4470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,e^2=gh,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, 8572114);