#8904541

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

4418 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8811737);