#147922

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

962 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)(5,6,10,16,12,17,11,34)(7,18,21,51,22,52,35,47)(8,14,26,44,27,46,36,45)(9,28,31,49,33,63,32,15)(13,29,38,58,40,64,39,20)(19,50,54,65,56,68,55,41)(23,59,60,66,53,67,57,37)(24,61,62,42,25,48,30,43);(5,7)(6,13)(8,23)(9,29)(10,21)(11,35)(12,22)(14,41)(15,47)(16,38)(17,40)(18,28)(19,25)(20,32)(24,56)(26,60)(27,53)(30,54)(31,58)(33,64)(34,39)(36,57)(37,43)(42,66)(44,50)(45,68)(46,65)(48,67)(49,51)(52,63)(55,62)(59,61);(2,4)(5,8)(6,14)(7,19)(9,30)(10,26)(11,36)(12,27)(13,37)(15,48)(16,44)(17,46)(18,50)(20,57)(21,54)(22,56)(23,29)(24,31)(25,32)(28,43)(33,62)(34,45)(35,55)(38,59)(39,67)(40,66)(41,47)(42,63)(49,61)(51,65)(52,68)(53,64)(58,60)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,f^b=fh,h^b=hi,d^c=dhi,f^c=fh,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, 55118);