#501265

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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	#61054	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

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

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, 408461);