#8760409

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

4651 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38);(1,2)(3,7)(4,5)(6,8)(10,15)(13,23)(16,19)(17,21)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39);(2,5)(6,7)(11,20)(12,22)(16,19)(17,21)(26,27)(28,35)(32,34)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,23)(20,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,18,15,24)(11,21,20,17)(12,19,22,16)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,f^2=gh,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, 8667605);