#7732004

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

4394 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=h,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, 7639200);