#347457

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63063	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2145 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)(9,10,16,22)(11,23,28,40)(12,29,33,43)(13,34,36,46)(14,21,15,20)(17,27,41,26)(18,31,44,32)(19,37,45,35)(24,49,52,38)(25,39,55,64)(30,42,60,66)(47,56,65,51)(48,53,61,54)(50,67,71,72)(57,59,62,58)(63,70,68,69);(3,7)(5,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,26)(16,28)(18,38)(19,39)(20,40)(21,23)(22,41)(29,47)(30,50)(31,56)(32,51)(33,52)(34,61)(35,54)(36,55)(37,53)(42,63)(43,65)(44,49)(45,64)(46,48)(57,67)(58,70)(59,69)(60,71)(62,72)(66,68);(1,2)(3,8)(4,6)(5,7)(9,12,16,33)(10,18,22,44)(11,24,28,52)(13,30,36,60)(14,31,15,32)(17,38,41,49)(19,42,45,66)(20,43,21,29)(23,47,40,65)(25,50,55,71)(26,51,27,56)(34,62,46,57)(35,58,37,59)(39,63,64,68)(48,67,61,72)(53,69,54,70);(9,13)(10,19)(11,25)(12,30)(14,35)(15,37)(16,36)(17,39)(18,42)(20,34)(21,46)(22,45)(23,48)(24,50)(26,53)(27,54)(28,55)(29,57)(31,58)(32,59)(33,60)(38,63)(40,61)(41,64)(43,62)(44,66)(47,67)(49,68)(51,69)(52,71)(56,70)(65,72)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254653);