#99122

order	512
nilpotency class	3
isoclinism	isoclinic class of #96924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

898 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 154 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)(11,15,16,13)(12,14)(19,22,23,24)(25,26,28,32,29,31,27,30);(2,5)(6,8)(17,18,21,20)(19,22,23,24)(25,27,29,28)(26,30,31,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,24)(21,23)(26,31)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=d,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,e^c=eh,f^c=fgh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6318);