#490437

order	512
nilpotency class	4
isoclinism	isoclinic class of #490434
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	C₄×C₈	C₂⁴
prime radical	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

469 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

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, 397633);