#99159

order	512
nilpotency class	3
isoclinism	isoclinic class of #99154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
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

686 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,12,16,13,15,11,14)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31);(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(10,15)(11,12)(17,20)(18,23,24,28)(19,26)(21,29)(22,30,25,31)(27,32)

Polycyclic presentation:

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

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, 6355);