#7634383

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

4867 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)(9,10,12,13)(11,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(2,5)(6,8)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(10,13)(14,15)(20,28)(23,32)(25,27)(29,31);(9,11,12,16)(10,14,13,15)(18,24)(22,30)(23,32)(29,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7541579);