#8724820

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

4600 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,23)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,6)(3,7)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,21,20,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(10,16)(11,18)(17,20)(19,24)(28,37)(32,36)(34,40)(38,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,16)(14,18)(17,21)(20,23)(22,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,12,14,22)(10,17,16,20)(11,19,18,24)(13,23,15,21)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40);(9,13,14,15)(10,18,16,11)(12,21,22,23)(17,19,20,24)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8632016);