#132045

order	512
nilpotency class	4
isoclinism	isoclinic class of #131970
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#537	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 740 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,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, 39241);