#135575

order	512
nilpotency class	4
isoclinism	isoclinic class of #135507
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=eg,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42771);