#6459111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366307);