#135586

order	512
nilpotency class	4
isoclinism	isoclinic class of #135524
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

542 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,f^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=egi,f^c=fh,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42782);