#394126

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3838 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:

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

Polycyclic presentation:

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

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, 301322);