#448809

order	512
nilpotency class	3
isoclinism	isoclinic class of #420424
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2699
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15921
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

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, 356005);