#504118

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411314);