#196313

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103509);