#497349

order	512
nilpotency class	4
isoclinism	isoclinic class of #497339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1091 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=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, 404545);