#133207

order	512
nilpotency class	4
isoclinism	isoclinic class of #133077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
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×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

900 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)(17,18)(19,26,28,23)(20,24,29,22)(21,25)(27,30)(31,32);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(18,22)(20,29)(23,30)(24,25)(26,31)(27,32);(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,26)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293030
maximal normal subgroup of	#72293030
maximal quotient of	#72293146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40403);