#499740

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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	#61057	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1258 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)(8,24,27,44)(9,28,32,42)(10,17,11,16)(13,35,38,21)(14,33,40,26)(15,30,43,31)(20,48,52,65)(22,53,34,41)(25,56,59,39)(29,45,60,37)(36,57,64,68)(46,51,67,50)(49,54,62,63)(55,61,66,58);(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)(25,57)(26,50)(27,52)(30,35)(32,60)(33,51)(39,65)(40,64)(42,47)(43,53)(44,63)(46,55)(48,56)(49,58)(59,68)(61,62)(66,67);(2,4)(5,8,12,27)(6,14,18,40)(7,20,23,52)(9,25,32,59)(10,33,11,26)(13,36,38,64)(15,39,43,56)(16,44,17,24)(19,46,47,67)(21,49,35,62)(22,51,34,50)(28,55,42,66)(29,57,60,68)(30,61,31,58)(37,63,45,54)(41,65,53,48);(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)(25,57)(26,50)(27,52)(30,35)(32,60)(33,51)(39,65)(40,64)(42,47)(43,53)(44,63)(46,55)(48,56)(49,58)(59,68)(61,62)(66,67)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,f^c=fi,e^d=eh,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, 406936);