#8914030

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4359 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8821226);