#93208

order	512
nilpotency class	5
isoclinism	isoclinic class of #93205
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₂²⋊C₈	C₂×C₄
Frattini, cosocle	#1780	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1780 ◁ G
LECS	G ▷ #1780 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 232 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) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fh,d^2=h,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, 404);