#501982

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28245	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(3,5)(4,6)(7,8)(9,10)(11,24)(12,31)(13,37)(14,23)(15,21)(16,22)(17,28)(18,35)(19,38)(20,25)(26,56)(27,51)(29,53)(30,44)(32,65)(33,64)(34,48)(36,54)(39,50)(40,70)(41,47)(42,62)(43,61)(45,60)(46,69)(49,68)(52,67)(55,59)(57,63)(58,71)(66,72);(1,3,4,7)(2,5,6,8)(9,11,25,30,15,29,14,28)(10,17,23,53,21,44,20,24)(12,32,36,69,35,66,34,57)(13,27,16,40,39,61,38,60)(18,46,54,65,31,63,48,72)(19,43,50,70,22,51,37,45)(26,41,62,68,59,67,58,33)(42,47,56,64,71,52,55,49);(2,6)(5,8)(9,12)(10,18)(11,26)(13,33)(14,34)(15,35)(16,41)(17,42)(19,47)(20,48)(21,31)(22,52)(23,54)(24,55)(25,36)(27,57)(28,58)(29,59)(30,62)(32,40)(37,49)(38,67)(39,68)(43,63)(44,71)(45,65)(46,51)(50,64)(53,56)(60,66)(61,69)(70,72);(9,13,14,38,15,39,25,16)(10,19,20,37,21,22,23,50)(11,27,28,60,29,61,30,40)(12,33,34,67,35,68,36,41)(17,43,24,45,44,51,53,70)(18,47,48,49,31,52,54,64)(26,57,58,66,59,69,62,32)(42,63,55,65,71,46,56,72)

Polycyclic presentation:

2^9|b^2=ef,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,g^c=gh,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, 409178);