#93202

order	512
nilpotency class	5
isoclinism	isoclinic class of #93191
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	C₂²×D₄	C₂×C₈
Frattini, cosocle	#1780	C₂²
prime radical	#3485	C₄
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1780 ◁ G
LECS	G ▷ #1780 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 282 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=d,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=dei,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, 398);