#502143

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
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

724 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:

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

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,e^b=ei,f^c=fh,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409339);