#249803

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60651	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1308 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156999);