#9005176

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

4532 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)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,37)(36,39)(38,40);(1,3,5,8)(2,6,7,4)(10,12)(15,17)(16,20)(22,23)(25,27)(26,28)(29,35)(30,31)(32,34)(33,36)(37,40)(38,39);(1,2)(3,4)(5,7)(6,8)(10,12)(11,18)(13,21)(22,23)(25,28,30,32)(26,31,34,27)(29,37,38,36)(33,35,40,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,27)(26,32)(28,34)(29,35)(30,31)(33,37)(36,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,38)(26,33,34,40)(27,35,31,39)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8912372);