#498813

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406009);