#502866

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

838 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410062);