#416216

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1258 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)(5,6,12,18,11,16,10,17)(7,19,24,37,23,13,22,43)(8,25,30,14,28,44,29,40)(9,31,34,41,33,42,32,15)(20,46,53,38,26,54,48,35)(21,49,52,45,51,61,50,36)(27,57,60,55,58,39,59,64)(47,63,68,67,56,62,66,65);(1,2,3,4)(5,7,10,24,11,23,12,22)(6,13,17,43,16,19,18,37)(8,26,29,48,28,20,30,53)(9,21,32,52,33,51,34,50)(14,38,25,54,40,35,44,46)(15,36,42,49,41,45,31,61)(27,56,59,66,58,47,60,68)(39,63,55,67,57,62,64,65);(5,8)(6,14)(7,20)(9,27)(10,30)(11,28)(12,29)(13,35)(15,39)(16,40)(17,44)(18,25)(19,38)(21,47)(22,53)(23,26)(24,48)(31,55)(32,60)(33,58)(34,59)(36,62)(37,46)(41,57)(42,64)(43,54)(45,63)(49,67)(50,68)(51,56)(52,66)(61,65);(5,9,12,34,11,33,10,32)(6,15,17,42,16,41,18,31)(7,21,22,50,23,51,24,52)(8,27,29,59,28,58,30,60)(13,36,43,49,19,45,37,61)(14,39,44,64,40,57,25,55)(20,47,53,68,26,56,48,66)(35,62,54,67,38,63,46,65)

Polycyclic presentation:

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

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, 323412);