#8424609

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

4352 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)(11,19,22,15)(12,23,21,16)(13,17,20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19)(26,30)(31,33)(32,37)(38,39);(1,3,5,6)(2,4,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,19)(12,16)(13,24)(14,18)(15,22)(17,20)(21,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,24,19,17);(1,4)(2,6)(3,7)(5,8)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=gh,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

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, 8331805);