#497343

order	512
nilpotency class	4
isoclinism	isoclinic class of #497333
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	#61090	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1101 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)(3,15,17,9,19,33,25,34)(4,11,21,41,23,42,26,39)(6,13,27,36,28,10,29,37)(16,44,47,31,48,38,20,52)(18,45,51,30,50,43,49,32)(22,40,56,61,55,35,54,60)(46,59,53,57,63,62,64,58);(1,3)(2,9)(4,20)(5,25)(6,18)(7,17)(8,19)(10,30)(11,38)(12,33)(13,32)(14,34)(15,24)(16,21)(22,53)(23,47)(26,48)(27,51)(28,50)(29,49)(31,39)(35,59)(36,45)(37,43)(40,62)(41,52)(42,44)(46,54)(55,64)(56,63)(57,60)(58,61);(2,10)(6,28)(9,30)(11,35)(12,36)(13,14)(15,43)(18,50)(22,55)(24,37)(27,29)(31,57)(32,34)(33,45)(38,59)(39,60)(40,42)(41,61)(44,62)(46,63)(49,51)(52,58)(53,64)(54,56);(1,4)(2,11)(3,16)(5,26)(6,22)(7,21)(8,23)(9,31)(10,35)(12,41)(13,40)(14,42)(15,44)(17,47)(18,46)(19,48)(20,25)(24,39)(27,56)(28,55)(29,54)(30,57)(32,58)(33,38)(34,52)(36,61)(37,60)(43,62)(45,59)(49,64)(50,63)(51,53)

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,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404539);