#149378

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56574);