#9057901

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

4743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965097);