#497345

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	#61091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1085 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,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, 404541);