#149328

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	#1824
prime radical	C₂⁴	C₂×Q₁₆
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,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, 56524);