#509291

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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

541 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)(3,4)(5,7)(6,8)(9,10)(11,16)(12,28)(13,17)(14,20)(15,21)(18,29)(19,33)(22,46)(23,51)(24,40)(25,36)(26,38)(27,39)(30,57)(31,43)(32,44)(34,41)(35,47)(37,52)(42,58)(45,56)(48,61)(49,62)(50,65)(53,71)(54,63)(55,64)(59,67)(60,68)(66,69)(70,72);(1,3)(2,6)(4,7)(5,8)(9,11,15,27)(10,16,21,39)(12,23,32,54)(13,24,33,26)(14,22,35,50)(17,40,19,38)(18,37,43,64)(20,46,47,65)(25,56,49,34)(28,51,44,63)(29,52,31,55)(30,48,60,70)(36,45,62,41)(42,66,67,71)(53,58,69,59)(57,61,68,72);(1,4,5,6)(2,3,7,8)(10,17)(11,22)(13,33)(14,34)(16,36)(18,28)(19,21)(20,47)(23,48)(24,49)(25,26)(27,50)(29,31)(30,59)(35,56)(37,61)(38,65)(39,62)(40,46)(42,67)(43,44)(51,66)(52,53)(54,70)(55,69)(58,60)(63,71)(64,72);(2,7)(3,8)(9,12,15,32)(10,18,21,43)(11,23,27,54)(13,31,33,29)(14,30,35,60)(16,37,39,64)(17,28,19,44)(20,42,47,67)(22,48,50,70)(24,55,26,52)(25,53,49,69)(34,59,56,58)(36,61,62,72)(38,63,40,51)(41,57,45,68)(46,66,65,71)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416487);