#7715497

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

4092 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)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,20)(23,24)(26,30)(28,36)(31,33)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23);(11,15)(12,18)(19,23)(20,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3,4,7)(2,5,6,8)(9,12,14,18)(10,15,17,11)(13,20,22,24)(16,23,21,19)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=h,e^2=g,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, 7622693);