#497328

order	512
nilpotency class	4
isoclinism	isoclinic class of #497323
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	#61054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1102 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)(7,29,30,13)(8,14)(9,16)(10,21,36,28)(17,43)(18,31,49,44)(19,41)(20,42)(22,51)(23,53,54,39)(24,40)(26,34)(27,38)(32,45)(33,46,57,55)(35,50,61,56)(37,48)(47,58,64,62)(52,59,63,60);(1,3,8,19)(2,9,14,34)(4,16,6,26)(5,22,24,45)(7,18,30,49)(10,31,36,44)(11,37,40,43)(12,41,25,15)(13,33,29,57)(17,27,48,20)(21,46,28,55)(23,52,54,63)(32,42,51,38)(35,60,61,59)(39,58,53,62)(47,56,64,50);(1,4,8,6)(2,10,14,36)(3,16,19,26)(5,20,24,27)(7,21,30,28)(9,31,34,44)(11,35,40,61)(12,29,25,13)(15,33,41,57)(17,45,48,22)(18,46,49,55)(23,50,54,56)(32,58,51,62)(37,60,43,59)(38,39,42,53)(47,63,64,52);(1,5,8,24)(2,11,14,40)(3,17,19,48)(4,20,6,27)(7,23,30,54)(9,32,34,51)(10,35,36,61)(12,42,25,38)(13,39,29,53)(15,43,41,37)(16,45,26,22)(18,47,49,64)(21,50,28,56)(31,58,44,62)(33,59,57,60)(46,63,55,52)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^c=gh,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, 404524);