#496613

order	512
nilpotency class	4
isoclinism	isoclinic class of #496611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21888
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

729 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)(3,16,21,40)(4,22)(5,28,20,38)(6,12)(7,32)(8,15)(9,19,39,34)(10,26)(11,29,46,55)(13,18)(14,31)(17,50,52,62)(23,59,54,42)(24,60,56,36)(25,57)(27,58)(30,48)(33,43)(35,47)(37,53)(41,51,49,63)(44,64)(45,61);(1,3,6,19,8,21,31,34)(2,9,14,40,15,39,12,16)(4,23,25,52,27,54,30,17)(5,18,29,35,20,53,55,7)(10,41,44,36,45,49,43,60)(11,37,38,47,46,13,28,32)(22,50,48,42,58,62,57,59)(24,33,51,61,56,64,63,26);(1,4,8,27)(2,10,15,45)(3,17,21,52)(5,24,20,56)(6,30,31,25)(7,33,35,64)(9,36,39,60)(11,42,46,59)(12,44,14,43)(13,48,37,57)(16,49,40,41)(18,26,53,61)(19,54,34,23)(22,47,58,32)(28,62,38,50)(29,63,55,51);(1,5)(2,11,15,46)(3,18)(4,24)(6,29)(7,34)(8,20)(9,37,39,13)(10,42,45,59)(12,28,14,38)(16,32,40,47)(17,26)(19,35)(21,53)(22,41,58,49)(23,33)(25,51)(27,56)(30,63)(31,55)(36,57,60,48)(43,50,44,62)(52,61)(54,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=ehi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,e^c=ei,f^c=fh,h^c=hi,f^d=fi,g^d=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, 403809);