#499912

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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	#60459	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1349 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)(7,19,23,41)(8,24,27,58)(9,17,11,15)(10,30,33,42)(13,28,37,21)(14,38,40,26)(16,29,44,32)(20,48,52,64)(22,54,56,36)(25,59,34,43)(31,45,61,39)(35,55,63,68)(46,65,62,50)(47,60,66,53)(49,57,51,67);(1,2)(3,4)(5,7)(6,13)(8,20)(9,28)(10,22)(11,21)(12,23)(14,35)(15,41)(16,36)(17,19)(18,37)(24,57)(25,51)(26,50)(27,52)(29,60)(30,47)(31,55)(32,53)(33,56)(34,49)(38,65)(39,62)(40,63)(42,66)(43,64)(44,54)(45,46)(48,59)(58,67)(61,68);(2,4)(5,8,12,27)(6,14,18,40)(7,20,23,52)(9,25,11,34)(10,31,33,61)(13,35,37,63)(15,39,17,45)(16,43,44,59)(19,46,41,62)(21,49,28,51)(22,55,56,68)(24,42,58,30)(26,29,38,32)(36,64,54,48)(47,57,66,67)(50,60,65,53);(5,7)(6,13)(8,20)(9,28)(10,22)(11,21)(12,23)(14,35)(15,41)(16,36)(17,19)(18,37)(24,57)(25,51)(26,50)(27,52)(29,60)(30,47)(31,55)(32,53)(33,56)(34,49)(38,65)(39,62)(40,63)(42,66)(43,64)(44,54)(45,46)(48,59)(58,67)(61,68)

Polycyclic presentation:

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

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, 407108);