#6813577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fhi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6720773);