#205689

order	512
nilpotency class	3
isoclinism	isoclinic class of #205553
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18,22,28,23,27,21,26)(19,24,30,38,31,40,36,39)(20,32,33,41,35,42,34,25)(29,43,46,37,45,47,44,48);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,22,36,23,31,21,30)(18,24,26,38,27,40,28,39)(20,29,33,44,35,45,34,46)(25,37,42,47,41,48,32,43);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,21,34,23,35,22,33)(18,25,28,32,27,41,26,42)(19,29,30,46,31,45,36,44)(24,37,39,43,40,48,38,47);(3,9)(7,14)(10,15)(12,16)(17,21,23,22)(18,26,27,28)(19,30,31,36)(20,33,35,34)(24,38,40,39)(25,32,41,42)(29,44,45,46)(37,43,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112885);