#130043

order	512
nilpotency class	4
isoclinism	isoclinic class of #130043
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8344
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#60714	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

628 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37239);