#148333

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	#1852
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 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)(5,6,10,16,12,17,11,33)(7,18,23,47,24,48,22,46)(8,14,28,42,29,43,27,41)(9,30,31,44,32,60,19,15)(13,34,37,55,38,49,36,21)(20,45,25,56,54,66,53,67)(26,58,59,65,51,64,50,40)(35,57,39,63,61,68,62,52);(5,7,19,49,11,22,32,55,12,24,31,34,10,23,9,21)(6,13,30,47,16,37,44,48,17,38,60,46,33,36,15,18)(8,25,50,52,27,20,51,68,29,53,59,63,28,54,26,57)(14,39,58,66,42,61,65,67,43,62,64,45,41,35,40,56);(2,4)(5,8)(6,14)(7,20)(9,26)(10,28)(11,27)(12,29)(13,35)(15,40)(16,42)(17,43)(18,45)(19,50)(21,52)(22,53)(23,25)(24,54)(30,58)(31,59)(32,51)(33,41)(34,57)(36,62)(37,39)(38,61)(44,65)(46,67)(47,56)(48,66)(49,68)(55,63)(60,64)

Polycyclic presentation:

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

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, 55529);