#394201

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2546 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=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, 301397);