#149147

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	#2065
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

420 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)(5,6,11,19,13,20,12,40)(7,21,26,42,28,51,27,60)(8,29,34,62,36,48,35,47)(9,17,10,38,37,18,31,16)(14,25,43,23,44,24,52,39)(15,45,49,65,50,32,61,33)(22,54,57,67,59,41,58,66)(30,63,55,68,56,53,64,46);(5,7)(6,14,20,44)(8,30)(9,25)(10,39)(11,27)(12,26)(13,28)(15,46,50,68)(16,42,38,60)(17,21,18,51)(19,52,40,43)(22,33)(23,31)(24,37)(29,54,48,41)(32,57)(34,64)(35,55)(36,56)(45,58)(47,67,62,66)(49,53,61,63)(59,65);(5,8,31,65,11,34,9,32,13,36,10,33,12,35,37,45)(6,15,17,29,40,61,16,47,20,50,18,48,19,49,38,62)(7,22,25,30,26,57,23,55,28,59,24,56,27,58,39,64)(14,41,42,46,52,67,21,53,44,54,60,68,43,66,51,63)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gi,h^b=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, 56343);