#6631468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 conjugacy classes of subgroups; only the 488 normal subgroups are shown.

normal subgroups only

This group has 1775 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538664);