#97312

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

418 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 96
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=g,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4508);