#7640878

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 7548074);