#7635333

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4849 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)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,31)(30,32);(2,5)(6,8)(11,15)(13,16);(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,26)(29,32);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(11,15)(13,16)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542529);