#491105

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,d^c=di,f^c=fh

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, 398301);