#133743

order	512
nilpotency class	4
isoclinism	isoclinic class of #133278
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

895 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 84 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)(17,18,21,24,22,25,20,30)(19,26,29,31,28,23,27,32);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(17,19,22,28)(18,23,25,26)(20,27,21,29)(24,31,30,32);(1,4)(2,7)(3,10)(5,13)(6,9)(8,12)(11,16)(14,15)(17,20)(18,24)(19,27)(21,22)(23,31)(25,30)(26,32)(28,29)

Polycyclic presentation:

2^9|a^2=f,b^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,e^b=eh,f^b=fg,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40939);