#7639625

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

4851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=g,f^2=gh,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, 7546821);