#7637916

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	#39883	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4695 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)(3,6)(4,5)(7,8)(17,18,22,25)(19,23,21,26)(20,28,29,24)(27,32,30,31);(2,5)(6,8)(10,14)(15,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(9,10)(11,15)(12,16)(13,14)(17,20,22,29)(18,24,25,28)(19,27,21,30)(23,31,26,32);(9,11,13,12)(10,15,14,16)(18,25)(23,26)(24,28)(31,32);(9,12,13,11)(10,15,14,16)(17,21,22,19)(18,26,25,23)(20,27,29,30)(24,31,28,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=hi,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, 7545112);