#7715543

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(9,10)(11,16)(12,14)(13,20)(15,18)(17,21)(19,22)(23,24)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,2,4,5)(3,6,7,8)(11,18)(15,16)(19,24)(22,23);(9,11,14,18)(10,15,12,16)(13,19,21,24)(17,22,20,23)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39);(9,13,14,21)(10,17,12,20)(11,19,18,24)(15,22,16,23)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622739);