#504788

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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)(3,7,8,6)(9,10,13,20,14,21,37,46,16,23,38,45,15,22,35,18)(11,24,27,42,28,52,60,44,29,53,58,43,25,40,34,17)(12,30,31,50,39,64,63,48,33,61,62,47,32,49,36,19)(26,41,55,65,54,68,71,70,57,67,72,69,56,66,59,51);(2,5)(6,7)(9,11,16,29)(10,17,23,44)(12,26,33,57)(13,34,38,60)(14,25,15,28)(18,24,46,53)(19,41,48,67)(20,40,45,52)(21,43,22,42)(27,37,58,35)(30,51,61,70)(31,59,62,71)(32,54,39,56)(36,55,63,72)(47,68,50,66)(49,65,64,69);(10,18)(11,25)(13,35)(14,15)(17,40)(19,30)(20,22)(21,45)(23,46)(24,43)(26,54)(27,58)(28,29)(31,36)(32,39)(37,38)(41,65)(42,53)(44,52)(47,64)(48,61)(49,50)(51,68)(56,57)(59,71)(62,63)(66,70)(67,69);(1,3,4,8)(2,6,5,7)(9,12)(10,19)(11,26)(13,36)(14,32)(15,39)(16,33)(17,41)(18,30)(20,49)(21,47)(22,50)(23,48)(24,51)(25,54)(27,59)(28,56)(29,57)(31,35)(34,55)(37,62)(38,63)(40,65)(42,66)(43,68)(44,67)(45,64)(46,61)(52,69)(53,70)(58,71)(60,72)

Polycyclic presentation:

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

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, 411984);