#494019

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#20723
prime radical	#3455	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

732 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)(3,13,14,37)(4,10)(5,11)(6,23,24,9)(7,12)(8,16,28,39)(15,35,38,58)(17,34)(18,31)(19,46,47,29)(20,32)(21,48,49,30)(22,33)(25,50)(26,41,53,59)(27,43)(36,40)(42,56)(44,62,63,54)(45,55)(51,64)(52,61)(57,60);(1,3)(2,8,12,28)(4,15)(5,16)(6,17)(7,14)(9,27,23,25)(10,26,32,53)(11,13,33,37)(18,41)(19,42)(20,38)(21,43)(22,39)(24,40)(29,52,46,51)(30,36,48,34)(31,35,55,58)(44,61)(45,59)(47,60)(49,50)(54,57,62,56)(63,64);(2,9)(3,14)(8,25)(10,29)(11,30)(12,23)(13,34)(15,38)(16,39)(17,40)(26,51)(27,28)(31,54)(32,46)(33,48)(35,56)(36,37)(41,59)(42,60)(43,50)(52,53)(55,62)(57,58)(61,64);(1,4,5,18,7,20,22,45)(2,10,11,31,12,32,33,55)(3,15,16,41,14,38,39,59)(6,19,21,44,24,47,49,63)(8,26,13,35,28,53,37,58)(9,29,30,54,23,46,48,62)(17,42,43,61,40,60,50,64)(25,51,34,56,27,52,36,57)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,g^b=gi,g^c=gh,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401215);