#93203

order	512
nilpotency class	5
isoclinism	isoclinic class of #93192
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂×C₄⋊C₄	C₂×C₈
Frattini, cosocle	#1818	C₂²
prime radical	#3493	C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1818 ◁ G
LECS	G ▷ #1818 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

208 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 26 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)(3,11,13,29,14,24,30,32)(4,15,17,27)(6,10,19,28)(7,20,21,16,22,31,26,12)(8,23,25,18)(33,34,37,43,39,44,58,76,40,45,59,77,38,60,63,75)(35,46,48,78,49,79,83,95,50,70,84,92,61,91,56,88)(36,51,53,85,54,42,69,71,55,72,57,89,62,73,87,96)(41,64,65,47,67,80,74,82,68,90,93,52,66,86,94,81);(1,3)(2,7)(4,16)(5,13)(6,14)(8,24)(9,26)(10,22)(11,15)(12,18)(17,31)(19,30)(20,23)(21,28)(25,29)(27,32)(33,35)(34,41)(36,52)(37,56)(38,61)(39,49)(40,50)(42,70)(43,74)(44,67)(45,68)(46,73)(47,55)(48,58)(51,79)(53,64)(54,81)(57,90)(59,83)(60,66)(62,82)(63,84)(65,75)(69,80)(71,95)(72,91)(76,93)(77,94)(78,85)(86,87)(88,96)(89,92)

Polycyclic presentation:

2^9|a^2=d,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=dehi,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,e^d=eh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399);