#9061451

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	#43117	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8968647);