#8930195

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

4331 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837391);