#7712576

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4468 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619772);