#137232

order	512
nilpotency class	4
isoclinism	isoclinic class of #137149
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,22,25,32,29,20,30,21,26,31,28,23,19,27,24);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,23,18,21,29,32,26)(20,30,22,28,27,24,31,25);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(20,27)(22,31)(24,30)(25,28)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=g,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265731, #72308481, #72308493
maximal normal subgroup of	#72265731, #72308481, #72308493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44428);