#93921

order	512
nilpotency class	5
isoclinism	isoclinic class of #93766
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1333
commutator	C₂×C₁₆	C₂×C₈
Frattini, cosocle	C₂×#346	C₂²
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

163 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(10,16,17,32,20,34,21,35)(11,22,25,36)(12,13)(14,30)(18,23,28,37,29,38,19,33)(24,31)(26,27,39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,23)(12,28)(13,29)(14,17)(15,20)(16,26)(18,31)(19,24)(21,30)(22,32)(25,38)(27,33)(34,39)(35,36)(37,40)

Polycyclic presentation:

2^9|a^2=d,c^2=fh,d^2=g,e^2=h,f^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=defhi,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227209
maximal normal subgroup of	#72227209
maximal quotient of	#72227205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1117);