#9712800

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	#51934	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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)(3,4)(5,8)(6,7)(9,10,16,18)(11,19,13,17)(12,22,14,15)(20,21,23,24)(28,36)(31,38)(35,40)(37,39);(1,3)(2,4)(5,6)(7,8)(9,11,16,13)(10,17,18,19)(12,20,14,23)(15,21,22,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2,7,6)(3,5,8,4)(9,12,16,14)(10,15,18,22)(11,20,13,23)(17,21,19,24)(25,27)(26,29)(28,35)(30,33)(31,37)(32,34)(36,40)(38,39);(1,4,7,5)(2,8,6,3)(9,13,16,11)(10,17,18,19)(12,20,14,23)(15,24,22,21)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,40)(34,38);(1,5,7,4)(2,8,6,3)(9,14,16,12)(10,15,18,22)(11,20,13,23)(17,24,19,21)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,40)(37,38);(1,6,7,2)(3,5,8,4)(9,15)(10,14)(11,21)(12,18)(13,24)(16,22)(17,23)(19,20)(25,29,30,32)(26,33,34,27)(28,37,36,39)(31,40,38,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,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, 9619996);