#93622

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂²	#1289
prime radical	C₂⁴	#106
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fg,e^2=fg,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 818);