#9223974

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

4613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131170);