#10247027

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

4489 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)(10,14)(12,20)(15,17)(18,21)(25,26,30,33)(27,34,29,31)(28,32,36,39)(35,40,37,38);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,17,19,15)(12,16,20,22)(18,24,21,23)(27,29)(28,36)(31,34)(32,39);(9,10,13,14)(11,15,19,17)(12,16,20,22)(18,23,21,24)(25,27,30,29)(26,31,33,34)(28,37,36,35)(32,40,39,38);(2,7)(3,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28,30,36)(26,32,33,39)(27,35,29,37)(31,38,34,40);(1,3)(2,4)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)(25,29,30,27)(26,31,33,34)(28,35,36,37)(32,40,39,38);(1,4)(2,8)(3,7)(5,6)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40)

Polycyclic presentation:

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

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, 10154223);