#99077

order	512
nilpotency class	3
isoclinism	isoclinic class of #98920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

454 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=ei,f^c=fhi

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, 6273);