#149138

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2101
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

414 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)(7,20,25,49)(8,27)(9,17)(10,16)(11,36)(12,19)(13,22,40,23)(14,29)(15,33)(18,35)(21,52,55,58)(24,50,57,41)(26,47,37,39)(28,61,63,42)(30,45)(31,44)(32,43)(34,48)(38,62,67,56)(46,65)(51,53,68,54)(59,64)(60,66);(1,2)(3,4)(5,7,11,37,12,25,35,26)(6,13,36,41,19,40,18,50)(8,28,31,54,32,63,66,53)(9,24,33,23,34,57,10,22)(14,42,59,68,46,61,45,51)(15,47,48,20,16,39,17,49)(21,29,62,64,55,65,56,30)(27,58,44,38,43,52,60,67);(2,4)(5,8,9,29,11,31,33,64,12,32,34,65,35,66,10,30)(6,14,15,43,18,45,17,44,19,46,16,27,36,59,48,60)(7,21,22,53,26,56,57,63,25,55,23,54,37,62,24,28)(13,38,20,51,41,52,39,42,40,67,49,68,50,58,47,61)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,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, 56334);