#148404

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#1862
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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)(5,6,10,17,12,18,11,35)(7,19,24,53,26,48,25,60)(8,14,29,44,30,46,36,45)(9,31,33,16,34,50,20,15)(13,37,39,23,40,54,52,22)(21,51,56,41,57,67,61,68)(27,62,59,66,55,65,58,38)(28,47,32,42,63,49,64,43);(5,7,20,54,11,25,34,23,12,26,33,37,10,24,9,22)(6,13,31,53,17,39,16,48,18,40,50,60,35,52,15,19)(8,27,32,57,29,59,63,61,30,55,64,21,36,58,28,56)(14,41,47,38,45,51,43,65,46,68,49,66,44,67,42,62);(2,4)(5,8)(6,14)(7,21)(9,32)(10,29)(11,36)(12,30)(13,38)(15,47)(16,49)(17,44)(18,46)(19,51)(20,28)(22,58)(23,59)(24,56)(25,61)(26,57)(27,37)(31,42)(33,63)(34,64)(35,45)(39,62)(40,66)(41,53)(43,50)(48,67)(52,65)(54,55)(60,68)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,d^c=dh,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, 55600);