#508444

order	512
nilpotency class	4
isoclinism	isoclinic class of #508310
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,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, 415640);