#6631467

order	512
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2842
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups; only the 488 normal subgroups are shown.

normal subgroups only

This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 488 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538663);