#8756030

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663226);