#93204

order	512
nilpotency class	5
isoclinism	isoclinic class of #93193
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×Q₈	C₂×C₈
Frattini, cosocle	#1780	C₂²
prime radical	#3487	C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1780 ◁ G
LECS	G ▷ #1780 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 279 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)(17,18,21,27,23,29,45,60,24,30,46,61,22,28,44,59)(19,31,33,63,35,64,67,80,36,54,68,79,34,69,42,74)(20,37,39,71,40,56,73,58,41,62,43,75,47,26,53,55)(25,48,49,32,51,65,77,66,52,76,78,38,50,72,57,70);(1,3)(2,6)(4,12)(5,11)(7,9)(8,14)(10,13)(15,16)(17,19)(18,25)(20,38)(21,42)(22,35)(23,34)(24,36)(26,54)(27,57)(28,51)(29,50)(30,52)(31,56)(32,41)(33,44)(37,64)(39,48)(40,66)(43,76)(45,68)(46,67)(47,70)(49,59)(53,65)(55,74)(58,80)(60,78)(61,77)(62,69)(63,75)(71,79)(72,73)

Polycyclic presentation:

2^9|a^2=d,c^2=i,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ci,d^b=de,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, 400);