#9712859

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	#50167	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620055);