#500336

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61261	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

728 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,30)(13,19)(14,39)(15,20)(16,23)(18,33)(21,37)(22,38)(24,57)(25,42)(26,51)(27,43)(28,55)(29,45)(31,64)(32,52)(34,56)(35,48)(36,54)(40,50)(41,59)(44,62)(46,58)(47,67)(49,68)(53,61)(60,72)(63,71)(65,70)(66,69);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,31)(13,25)(14,27)(15,26)(16,29)(18,46)(19,42)(20,51)(21,53)(22,55)(23,45)(24,34)(28,38)(30,64)(32,65)(33,58)(35,59)(36,66)(37,61)(39,43)(40,62)(41,48)(44,50)(47,72)(49,71)(52,70)(54,69)(56,57)(60,67)(63,68);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,24)(13,32)(14,34)(15,33)(16,36)(17,41)(19,47)(20,52)(21,54)(22,56)(23,49)(25,58)(26,31)(27,59)(28,63)(29,60)(30,39)(35,38)(37,67)(40,68)(42,64)(43,71)(44,72)(45,70)(46,53)(48,50)(51,57)(55,69)(61,65)(62,66);(2,7)(6,8)(9,13,16,38)(10,19,23,22)(11,25,29,28)(12,32,36,35)(14,15,37,40)(17,42,45,55)(18,47,49,56)(20,21,50,39)(24,58,60,63)(26,61,62,27)(30,52,54,48)(31,65,66,59)(33,67,68,34)(41,64,70,69)(43,51,53,44)(46,72,71,57)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,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, 407532);