#412017

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1312 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)(7,13,21,35)(8,14,26,40)(9,27,30,41)(10,17,11,16)(15,28,42,29)(19,45,47,33)(20,49,50,63)(22,43,31,36)(23,37,48,59)(24,53,55,64)(25,44,32,39)(34,56,60,51)(38,57,62,54)(46,61,66,68)(52,58,67,65);(1,2)(3,4)(5,7)(6,13)(8,23)(9,20)(10,31)(11,22)(12,21)(14,37)(15,34)(16,43)(17,36)(18,35)(19,25)(24,52)(26,48)(27,49)(28,56)(29,51)(30,50)(32,47)(33,39)(38,61)(40,59)(41,63)(42,60)(44,45)(46,54)(53,58)(55,67)(57,66)(62,68)(64,65);(1,3,4,2)(5,8)(6,14)(7,19)(9,24)(10,32)(11,25)(12,26)(13,33)(15,38)(16,44)(17,39)(18,40)(20,46)(21,47)(22,48)(23,31)(27,53)(28,57)(29,54)(30,55)(34,58)(35,45)(36,59)(37,43)(41,64)(42,62)(49,68)(50,66)(51,67)(52,56)(60,65)(61,63);(5,9,10,28,12,30,11,29)(6,15,16,41,18,42,17,27)(7,20,31,56,21,50,22,51)(8,24,32,57,26,55,25,54)(13,34,43,63,35,60,36,49)(14,38,44,64,40,62,39,53)(19,46,23,52,47,66,48,67)(33,58,37,61,45,65,59,68)

Polycyclic presentation:

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

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, 319213);