#6813450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(19,26)(23,31)(27,32)(28,29);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,18)(19,27)(20,24)(21,25)(22,30)(23,28)(26,32)(29,31);(10,14)(13,16)(17,19,22,28,21,26,20,29)(18,23,24,27,25,31,30,32);(2,5)(6,8)(17,20,21,22)(18,24,25,30)(19,28,26,29)(23,32,31,27);(1,3,4,7)(2,6,5,8)(17,19,20,29,21,26,22,28)(18,23,30,32,25,31,24,27)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6720646);