#8925016

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

4602 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:

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

Polycyclic presentation:

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

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, 8832212);