#7636326

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	#38868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4719 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)(3,7)(4,6)(5,8)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3,5,4)(2,6,8,7)(10,13)(14,16);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(18,23)(22,25)(24,30)(29,31);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(19,26)(22,25)(27,32)(29,31);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^2=g,f^2=h,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, 7543522);