#133760

order	512
nilpotency class	4
isoclinism	isoclinic class of #133293
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×C₄⋊C₄	C₂³
prime radical	#36721	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

897 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 897 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,11,13,16)(4,9,5,8)(6,10)(7,12)(14,15)(17,18,20,25,21,23,28,31)(19,24,27,32,26,22,29,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(18,22)(19,26)(23,24)(25,32)(27,29)(30,31);(1,4)(2,8)(3,12)(5,6)(7,11)(9,10)(13,14)(15,16)(18,23)(20,28)(22,24)(27,29)

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,f^c=fi,f^d=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, 40956);