#497338

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

1105 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)(3,15)(4,12)(5,11)(6,25)(7,13,29,42)(8,14)(9,16)(10,21,35,28)(17,44)(18,30,50,57)(19,40)(20,41)(22,52)(23,38,55,59)(24,39)(26,33)(27,37)(31,45)(32,49,43,46)(34,51,60,54)(36,48)(47,56,64,61)(53,62,63,58);(1,3,8,19)(2,9,14,33)(4,16,6,26)(5,22,24,45)(7,18,29,50)(10,30,35,57)(11,36,39,44)(12,40,25,15)(13,32,42,43)(17,27,48,20)(21,46,28,49)(23,53,55,63)(31,41,52,37)(34,58,60,62)(38,61,59,56)(47,54,64,51);(1,4,8,6)(2,10,14,35)(3,16,19,26)(5,20,24,27)(7,28,29,21)(9,30,33,57)(11,34,39,60)(12,13,25,42)(15,43,40,32)(17,45,48,22)(18,49,50,46)(23,54,55,51)(31,56,52,61)(36,58,44,62)(37,59,41,38)(47,53,64,63);(1,5,8,24)(2,11,14,39)(3,17,19,48)(4,20,6,27)(7,23,29,55)(9,31,33,52)(10,34,35,60)(12,41,25,37)(13,38,42,59)(15,44,40,36)(16,45,26,22)(18,47,50,64)(21,51,28,54)(30,56,57,61)(32,58,43,62)(46,63,49,53)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=i,f^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, 404534);