#148232

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55428);