#135496

order	512
nilpotency class	4
isoclinism	isoclinic class of #135477
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 695 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,e^c=ei,f^c=fgh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42692);