#417614

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2309 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 324810);