#196316

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#525	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

(1,2,5,7,4,3,6,8)(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);(1,3,5,8,4,2,6,7)(9,11)(10,15,18,28)(12,19)(13,20)(14,21)(16,26,22,34)(17,27,30,38)(23,31)(24,32)(25,33)(29,37,35,40)(36,39);(2,3)(7,8)(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);(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=eg,d^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,e^b=eh,f^b=fh,g^b=gh,e^c=eh,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, 103512);