#501250

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61057	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408446);