#205952

order	512
nilpotency class	3
isoclinism	isoclinic class of #205769
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113148);