#135472

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	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

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

normal subgroups only

This group has 625 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)(3,12,14,7)(4,16,17,10)(5,9)(6,11)(8,13,22,27)(15,26,30,20)(18,31,32,24)(19,25)(21,28,23,29);(1,3,6,15)(2,7,11,20)(4,13,17,29)(5,14,19,30)(8,16,23,10)(9,12,25,26)(18,28,32,27)(21,31,22,24);(1,3)(2,8,9,21)(4,13)(5,14)(6,15)(7,10,12,24)(11,23,25,22)(16,26,31,20)(17,29)(18,28)(19,30)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fgh,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, 42668);