#148567

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,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, 55763);