#457608

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16298
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1482 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:

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

Polycyclic presentation:

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

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, 364804);