#8101214

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

4855 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8008410);