#8271924

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

4715 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8179120);