#8807541

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
prime radical	#51531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=gi,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, 8714737);