#501077

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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	#60459	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ 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:

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

Polycyclic presentation:

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

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, 408273);