#430411

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1165 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)(5,6,12,18,11,17,10,16)(7,19,24,13,23,39,22,38)(8,25,35,42,29,14,28,43)(9,30,34,15,33,46,32,45)(20,47,50,62,49,40,26,36)(21,37,52,64,51,63,31,44)(27,55,59,53,58,60,57,41)(48,66,68,65,56,61,54,67);(1,3,4,2)(5,7)(6,13)(8,26)(9,31)(10,24)(11,23)(12,22)(14,40)(15,44)(16,39)(17,38)(18,19)(20,28)(21,32)(25,47)(27,56)(29,50)(30,37)(33,52)(34,51)(35,49)(36,42)(41,66)(43,62)(45,64)(46,63)(48,58)(53,61)(54,57)(55,67)(59,68)(60,65);(2,3)(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,35)(13,36)(15,41)(16,42)(17,25)(18,43)(19,40)(21,48)(22,26)(23,49)(24,50)(30,60)(31,54)(32,57)(33,58)(34,59)(37,61)(38,62)(39,47)(44,65)(45,53)(46,55)(51,56)(52,68)(63,66)(64,67);(2,3)(5,9,10,32,11,33,12,34)(6,15,16,30,17,45,18,46)(7,21,22,31,23,51,24,52)(8,27,28,57,29,58,35,59)(13,37,19,44,38,63,39,64)(14,41,42,60,25,53,43,55)(20,48,26,54,49,56,50,68)(36,61,40,65,62,66,47,67)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,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, 337607);