#7642158

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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)(12,20)(16,21)(18,24)(22,23)(25,26,32,35)(27,31,29,33)(28,37,30,34)(36,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,32,29)(26,33,35,31)(28,36,30,39)(34,40,37,38);(10,14)(15,17)(16,21)(22,23)(25,28,32,30)(26,34,35,37)(27,36,29,39)(31,38,33,40);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,32,27)(26,31,35,33)(28,36,30,39)(34,40,37,38);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,30,32,28)(26,34,35,37)(27,36,29,39)(31,40,33,38);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,31)(26,27)(28,38)(29,35)(30,40)(32,33)(34,36)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

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, 7549354);