#132957

order	512
nilpotency class	4
isoclinism	isoclinic class of #132531
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#43882	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

596 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 74 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)(4,8,14,7)(5,9,12,16)(6,10)(13,15)(17,18,20,25,21,22,28,30)(19,23,27,31,26,24,29,32);(2,7)(4,14)(5,12)(8,10)(9,11)(15,16)(18,22)(19,26)(23,24)(27,29);(1,3,6,13)(2,7)(4,12,14,5)(8,10)(9,11)(15,16)(18,23,22,24)(20,28)(25,31,30,32)(27,29)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40153);