#448651

order	512
nilpotency class	3
isoclinism	isoclinic class of #448650
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15914
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

921 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,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, 355847);