#10040833

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9948029);