#449962

order	512
nilpotency class	3
isoclinism	isoclinic class of #449961
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15954
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

993 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)(3,13)(4,18,22,9)(5,10)(6,26)(7,11)(8,16)(12,28)(14,39,41,29)(15,35)(17,31)(19,32,47,43)(20,33,49,44)(21,34,51,45)(23,30)(24,37)(25,55)(27,38)(36,54)(40,56,52,61)(42,57)(46,59,48,58)(50,63,64,60)(53,62);(1,3)(2,8,11,13)(4,19)(5,23,24,15)(6,27)(7,16)(9,32,18,43)(10,35)(12,38,26,31)(14,21)(17,28)(20,48,49,46)(22,47)(25,42,54,53)(29,45,39,34)(30,37)(33,59)(36,57)(40,50,52,64)(41,51)(44,58)(55,62)(56,63)(60,61);(1,4)(2,9)(3,14,16,41)(5,20)(6,21)(7,22)(8,29,13,39)(10,33)(11,18)(12,34)(15,40,23,52)(17,19,27,47)(24,49)(25,50)(26,45)(28,51)(30,56,35,61)(31,32,38,43)(36,60)(37,44)(42,46,53,48)(54,64)(55,63)(57,58,62,59);(1,5,7,24)(2,10,11,37)(3,15,16,23)(4,20,22,49)(6,25,28,54)(8,30,13,35)(9,33,18,44)(12,36,26,55)(14,40,41,52)(17,42,27,53)(19,46,47,48)(21,50,51,64)(29,56,39,61)(31,57,38,62)(32,58,43,59)(34,60,45,63)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357158);