#7768033

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7675229);