#196203

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
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	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103399);