#196181

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1263 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:

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

Polycyclic presentation:

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

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, 103377);