#495849

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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)(5,6,11,17)(7,19,23,38)(8,25)(9,18,12,15)(10,31,33,43)(13,24,37,21)(14,39)(16,30,45,34)(20,48)(22,36,55,65)(26,58)(27,44)(28,42)(29,59)(32,40)(35,54)(41,60)(46,66)(47,53,61,56)(49,67)(50,57)(51,62)(52,64)(63,68);(1,2,3,4)(5,7,12,24,11,23,9,21)(6,13,18,38,17,37,15,19)(8,20,26,49,29,52,27,50)(10,22,34,56,33,55,30,53)(14,35,40,62,42,63,41,46)(16,36,31,61,45,65,43,47)(25,57,44,64,59,67,58,48)(28,51,32,54,39,66,60,68);(5,8)(6,14)(7,20)(9,27)(10,32)(11,29)(12,26)(13,35)(15,41)(16,44)(17,42)(18,40)(19,46)(21,50)(22,54)(23,52)(24,49)(25,43)(28,30)(31,59)(33,60)(34,39)(36,64)(37,63)(38,62)(45,58)(47,57)(48,65)(51,53)(55,68)(56,66)(61,67);(8,26,29,27)(10,33)(14,40,42,41)(16,45)(20,49,52,50)(22,55)(25,58,59,44)(28,60,39,32)(30,34)(31,43)(35,62,63,46)(36,65)(47,61)(48,67,64,57)(51,68,66,54)(53,56)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,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, 403045);