#128235

order	512
nilpotency class	3
isoclinism	isoclinic class of #108504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#376
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂³	#433
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 96 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=di,e^b=eg,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35431);