#502147

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,10,17,37,16,36,52,12,20,11,18,9,19,35,53)(7,21,24,54,23,13,39,41,27,45,25,44,26,42,38,14)(8,28,31,62,30,15,46,48,34,51,32,63,33,50,64,49)(22,40,57,67,59,60,66,68,29,47,58,43,56,55,65,61);(1,2)(3,4)(5,7,10,24,37,23,36,39,12,27,11,25,9,26,35,38)(6,13,17,41,16,45,52,44,20,42,18,14,19,21,53,54)(8,29,31,58,30,56,46,65,34,22,32,57,33,59,64,66)(15,47,48,43,51,55,63,61,50,40,49,67,28,60,62,68);(2,4)(5,7,12,27)(6,14,20,41)(8,22,34,29)(9,23,37,26)(10,38,11,39)(13,17,42,18)(15,43,50,67)(16,44,19,54)(21,52,45,53)(24,36,25,35)(28,61,51,68)(30,56,33,59)(31,65,32,66)(40,48,47,49)(46,58,64,57)(55,63,60,62);(5,8)(6,15)(7,22)(9,33)(10,31)(11,32)(12,34)(13,40)(14,43)(16,51)(17,48)(18,49)(19,28)(20,50)(21,55)(23,59)(24,57)(25,58)(26,56)(27,29)(30,37)(35,64)(36,46)(38,65)(39,66)(41,67)(42,47)(44,68)(45,60)(52,63)(53,62)(54,61)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409343);