#134468

order	512
nilpotency class	4
isoclinism	isoclinic class of #124826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=i,e^2=h,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,f^b=fi,e^c=eh,f^c=fgh,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41664);