#8678962

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gh,f^2=gi,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, 8586158);