#148735

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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)(7,18,22,37)(8,23)(9,14)(10,31)(11,17)(12,20,38,53)(13,26)(15,29)(16,30)(19,48,51,55)(21,36,32,43)(24,57,49,68)(25,44)(27,61)(28,56)(33,46)(34,42)(35,59,64,54)(39,67,66,50)(40,58)(41,60)(45,63)(47,62,65,52);(1,2)(3,4)(5,7,11,22)(6,12)(8,24)(9,20,30,53)(10,32,34,21)(13,39,41,66)(14,18)(15,43)(16,37)(17,38)(19,25)(23,55,56,48)(26,59)(27,51)(28,49)(29,62,63,52)(31,47)(33,50)(35,40,64,46)(36,45)(42,65)(44,57,61,68)(54,60)(58,67);(2,4)(5,8,9,25,11,28,30,27)(6,13,14,40,17,41,16,46)(7,19,20,49,22,51,53,24)(10,33,29,26,34,58,63,60)(12,35,18,39,38,64,37,66)(15,44,42,56,45,61,31,23)(21,54,52,50,32,59,62,67)(36,48,47,68,43,55,65,57)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=eh,f^c=fi,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, 55931);