#9175004

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

4722 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9082200);