#9368154

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9275350);