#148538

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei

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, 55734);