#8810152

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

4605 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)(4,8)(5,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,31)(27,35,36,32)(28,33,38,37)(29,34,39,40);(1,2,5,6)(3,7,8,4)(10,14)(12,21)(15,17)(19,24)(26,31)(29,39)(32,35)(33,37);(1,3)(2,4)(5,8)(6,7)(10,14)(11,18)(16,20)(19,24)(27,36)(28,38)(32,35)(33,37);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,27,30,36)(26,32,31,35)(28,34,38,40)(29,37,39,33);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,30,38)(26,33,31,37)(27,34,36,40)(29,35,39,32);(1,4)(2,3)(5,7)(6,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,34)(27,37)(28,35)(30,39)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717348);