#9838170

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9745366);