#6453405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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)(11,14)(12,15)(13,16)(17,18,23,27)(19,22,28,24)(20,25,30,32)(21,26,31,29);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,25)(21,26)(22,27)(23,28)(29,31)(30,32);(9,11,13,12)(10,15,16,14)(17,20)(18,25)(19,21)(22,26)(23,30)(24,29)(27,32)(28,31);(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,20)(22,25)(23,31)(24,32)(27,29)(28,30);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,19,27,28)(20,29,30,26)(21,25,31,32)

Polycyclic presentation:

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

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, 6360601);