#133157

order	512
nilpotency class	4
isoclinism	isoclinic class of #133089
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#36957	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,10,12,7)(4,8)(5,9)(6,11,15,13)(14,16)(17,18)(19,25,27,31)(20,28,29,23)(21,24)(22,26)(30,32);(1,3)(2,6)(4,13)(5,12)(7,8)(9,15)(10,16)(11,14)(17,19)(18,22,24,30)(20,26)(21,27)(23,25,28,31)(29,32);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,14)(15,16)(20,29)(23,28)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fh,f^d=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, 40353);