#512297

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419493);