#9040827

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	#35740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,13)(14,15)(16,18)(19,20)(21,23)(22,24)(25,26,31,35)(27,32,30,34)(28,38,29,33)(36,40,37,39);(2,5)(6,7)(10,15)(11,18)(12,21)(20,24)(25,27,31,30)(26,32,35,34)(28,37,29,36)(33,40,38,39);(1,2)(3,6)(4,5)(7,8)(12,21)(13,23)(19,22)(20,24)(25,28,31,29)(26,33,35,38)(27,36,30,37)(32,39,34,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,21)(23,24)(25,29,31,28)(26,33,35,38)(27,37,30,36)(32,39,34,40);(9,12)(10,13)(11,19)(14,21)(15,23)(16,24)(17,20)(18,22)(25,30,31,27)(26,34,35,32)(28,36,29,37)(33,39,38,40);(9,13)(10,12)(11,20)(14,23)(15,21)(16,22)(17,19)(18,24)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8948023);