#97315

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

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

normal subgroups only

This group has 416 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);(1,3,7,15)(2,8,10,12)(4,13,18,31)(5,14,6,21)(9,22,25,27)(11,23,19,24)(16,32,26,28)(17,29,20,30)(33,35)(34,40)(36,45)(37,52)(38,46)(39,47)(41,54)(42,55)(43,56)(44,51)(48,59)(49,62)(50,63)(53,61)(57,64)(58,60)(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)(2,9)(3,13)(5,20)(6,17)(7,18)(8,22)(10,25)(11,26)(12,27)(14,30)(15,31)(16,19)(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)(65,68,70,81)(66,72,74,79)(67,76,78,93)(69,83,84,80)(71,85,86,95)(73,89,90,87)(75,91,88,96)(77,94,82,92)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,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, 4511);