#7732816

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

4375 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7640012);