#7640482

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

4934 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)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31);(2,5)(6,8)(10,13)(11,15)(17,19)(18,23)(20,30)(21,27)(22,28)(24,26)(25,32)(29,31);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,30)(22,28)(23,26)(25,32)(29,31);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,25,26,29)(19,27,28,30)(23,31,24,32);(10,13)(14,16)(17,21,22,20)(18,25,26,29)(19,27,28,30)(23,32,24,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|d^2=hi,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, 7547678);