#97083

order	512
nilpotency class	3
isoclinism	isoclinic class of #97080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1242 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1242 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fg,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4279);