#9947337

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	#37529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4238 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)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,26,30,33)(27,31,35,28)(29,37,39,32)(34,36,40,38);(1,3)(2,4)(5,6)(7,8)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,2,5,8)(3,7,6,4)(25,27,30,35)(26,28,33,31)(29,34,39,40)(32,36,37,38);(1,4)(2,6)(3,8)(5,7)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,40,37,34);(1,3)(2,7)(4,8)(5,6)(9,14,15,11)(10,16,12,18)(13,19,22,21)(17,24,20,23)(25,29,30,39)(26,32,33,37)(27,34,35,40)(28,36,31,38)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854533);