#267886

order	512
nilpotency class	3
isoclinism	isoclinic class of #267851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

891 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,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, 175082);