#93210

order	512
nilpotency class	5
isoclinism	isoclinic class of #93205
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₂²⋊C₈	C₂×C₄
Frattini, cosocle	#1818	C₂²
prime radical	C₂⁴	#104
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1818 ◁ G
LECS	G ▷ #1818 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

163 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 96
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 96 points:

(1,2,5,9,6,10,18,26)(3,11,13,27,14,23,30,32)(4,15,16,31,17,8,22,24)(7,19,20,12,21,28,25,29)(33,34,37,42)(35,45,47,75)(36,50,53,70)(38,44,58,71)(39,43,57,72)(40,60,61,84)(41,64,67,55)(46,63,78,69)(48,65,81,93)(49,76,80,90)(51,62,83,91)(52,66,85,89)(54,74,86,96)(56,87,59,82)(68,88,94,77)(73,92,95,79);(1,3)(2,7)(4,12)(5,13)(6,14)(8,23)(9,25)(10,21)(11,15)(16,19)(17,29)(18,30)(20,26)(22,28)(24,32)(27,31)(33,35,39,49)(34,40,43,62)(36,51,55,84)(37,56,57,81)(38,59,52,48)(41,65,50,82)(42,69,72,95)(44,73,66,63)(45,74,76,68)(46,77,79,54)(47,58,80,85)(53,78,64,92)(60,88,83,86)(61,89,91,71)(67,90,70,75)(87,94,93,96)

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fh,d^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,c^b=cf,d^b=def,f^b=fh,g^b=gi,d^c=dg,e^c=ei,e^d=eh

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406);