#499164

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2054
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1009 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)(3,16)(4,22)(5,29)(6,32)(7,12)(8,34)(9,15)(10,18)(11,26)(13,31)(14,30)(17,36)(19,46)(20,38)(21,45)(23,47)(24,52)(25,57)(27,42)(28,58)(33,40)(35,37)(39,51)(41,54)(43,56)(44,55)(48,60)(49,61)(50,63)(53,64)(59,62);(1,3)(2,10)(4,23)(5,19)(6,33)(7,18)(8,35)(9,21)(11,41)(12,37)(13,45)(14,46)(15,40)(16,29)(17,27)(20,30)(22,36)(24,53)(25,50)(26,48)(28,49)(31,51)(32,38)(34,39)(42,61)(43,63)(44,64)(47,57)(52,60)(54,56)(55,59)(58,62);(1,4,5,25,30,56,6,26,9,24,31,55,8,28,7,27)(2,11,12,43,34,57,13,22,15,42,32,58,14,44,29,52)(3,17,18,49,35,59,51,53,21,48,33,54,20,50,19,23)(10,36,16,47,46,63,38,41,40,60,45,64,39,62,37,61);(4,24)(11,42)(17,48)(22,52)(23,53)(25,55)(26,27)(28,56)(36,60)(41,61)(43,58)(44,57)(47,64)(49,54)(50,59)(62,63)

Polycyclic presentation:

2^9|c^2=e,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307185, #72307209, #72307277, …
maximal normal subgroup of	#72307185, #72307209, #72307277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406360);