#99329

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂³	#385
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

374 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dg,e^b=eh,d^c=dh,e^c=egh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6525);