#455394

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16263
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362590);