#7731101

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

4435 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=i,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, 7638297);