#6357082

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2068 conjugacy classes of subgroups; only the 638 normal subgroups are shown.

normal subgroups only

This group has 2068 conjugacy classes of subgroups — too many to draw. Only its 638 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225180, #72225185, #72225248, …
maximal normal subgroup of	#72225180, #72225185, #72225248, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264278);