#427888

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

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, 335084);