#7634883

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
prime radical	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4958 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,26)(22,25)(24,29)(27,32)(30,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(10,14)(13,16)(18,25)(20,29)(23,26)(27,30);(18,25)(23,26)(24,28)(31,32);(17,21,22,19)(18,26,25,23)(20,30,29,27)(24,32,28,31)

Polycyclic presentation:

2^9|b^2=h,c^2=g,f^2=hi,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, 7542079);