#394093

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3846 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)(7,19,24,35)(8,14,29,41)(9,15,31,43)(10,17,11,16)(13,23,36,22)(20,37,48,58)(21,45,50,61)(25,33,47,59)(26,53,55,38)(27,40,28,39)(30,44,32,42)(34,51,60,49)(46,65,67,62)(52,68,66,57)(54,63,56,64);(2,4)(5,7,12,24)(6,13,18,36)(8,25,29,47)(9,21,31,50)(10,23,11,22)(14,37,41,58)(15,34,43,60)(16,35,17,19)(20,27,48,28)(26,52,55,66)(30,51,32,49)(33,39,59,40)(38,62,53,65)(42,61,44,45)(46,54,67,56)(57,63,68,64);(5,8)(6,14)(7,20)(9,26)(10,28)(11,27)(12,29)(13,33)(15,38)(16,40)(17,39)(18,41)(19,37)(21,46)(22,25)(23,47)(24,48)(30,56)(31,55)(32,54)(34,57)(35,58)(36,59)(42,64)(43,53)(44,63)(45,62)(49,52)(50,67)(51,66)(60,68)(61,65);(5,9)(6,15)(7,21)(8,26)(10,30)(11,32)(12,31)(13,34)(14,38)(16,42)(17,44)(18,43)(19,45)(20,46)(22,49)(23,51)(24,50)(25,52)(27,54)(28,56)(29,55)(33,57)(35,61)(36,60)(37,62)(39,63)(40,64)(41,53)(47,66)(48,67)(58,65)(59,68)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi,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, 301289);