#403106

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1172 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)(5,6,12,16,10,18,11,17)(7,19,23,36,22,13,31,37)(8,14,32,40,26,42,27,41)(9,28,30,43,29,44,33,15)(20,34,24,53,48,45,52,38)(21,49,59,46,50,62,51,35)(25,55,57,64,56,58,60,39)(47,66,68,61,67,63,54,65);(5,7)(6,13,18,19)(8,24)(9,21)(10,22)(11,31)(12,23)(14,38,42,53)(15,35,43,46)(16,37,17,36)(20,27)(25,54)(26,52)(28,49,44,62)(29,50)(30,59)(32,48)(33,51)(34,41,45,40)(39,63,64,66)(47,57)(55,65,58,61)(56,68)(60,67);(5,8,10,26)(6,14,18,42)(7,20,22,48)(9,25,29,56)(11,32,12,27)(13,34,19,45)(15,39,43,64)(16,41,17,40)(21,47,50,67)(23,52,31,24)(28,58,44,55)(30,60,33,57)(35,61,46,65)(36,53,37,38)(49,66,62,63)(51,68,59,54);(2,4)(5,9,10,29)(6,15)(7,21,22,50)(8,25,26,56)(11,30,12,33)(13,35)(14,39)(16,44)(17,28)(18,43)(19,46)(20,47,48,67)(23,51,31,59)(24,54,52,68)(27,57,32,60)(34,61)(36,49)(37,62)(38,63)(40,58)(41,55)(42,64)(45,65)(53,66)

Polycyclic presentation:

2^9|a^2=fg,b^2=efi,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 310302);