#135488

order	512
nilpotency class	4
isoclinism	isoclinic class of #135471
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 528 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fghi,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, 42684);