#412088

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1300 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)(7,19,24,37)(8,25,29,42)(9,31,34,43)(10,17,11,16)(13,23,38,22)(14,30,41,28)(15,32,44,33)(20,35,26,39)(21,50,53,36)(27,40,57,64)(45,48,61,49)(46,52,62,51)(47,65,55,68)(54,56,59,58)(60,67,63,66);(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,39)(15,36)(16,37)(17,19)(18,38)(20,29)(25,45)(27,55)(28,48)(30,49)(31,46)(32,52)(33,51)(34,53)(35,41)(40,63)(42,61)(43,62)(44,50)(47,57)(54,65)(56,66)(58,67)(59,68)(60,64);(2,4)(5,8,10,28,12,29,11,30)(6,14,16,25,18,41,17,42)(7,20,22,48,24,26,23,49)(9,27,32,56,34,57,33,58)(13,35,19,45,38,39,37,61)(15,40,43,54,44,64,31,59)(21,47,51,66,53,55,52,67)(36,60,46,65,50,63,62,68);(5,9,10,32,12,34,11,33)(6,15,16,43,18,44,17,31)(7,21,23,52,24,53,22,51)(8,27,28,56,29,57,30,58)(13,36,37,62,38,50,19,46)(14,40,25,54,41,64,42,59)(20,47,49,67,26,55,48,66)(35,60,61,68,39,63,45,65)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,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, 319284);