#347886

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 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:

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

Polycyclic presentation:

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

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, 255082);