#149352

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×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=gh,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, 56548);