#244545

order	512
nilpotency class	3
isoclinism	isoclinic class of #244523
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	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151741);