#8724843

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

4596 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)(9,10,14,16)(11,15,19,22)(12,21,23,17)(13,18,24,20)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,6)(3,7)(9,11,14,19)(10,15,16,22)(12,20,23,18)(13,17,24,21)(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)(11,19)(17,21)(18,20)(28,37)(32,36)(34,40)(38,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,22)(17,24)(18,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,12,14,23)(10,17,16,21)(11,20,19,18)(13,15,24,22)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40);(9,13,14,24)(10,18,16,20)(11,17,19,21)(12,22,23,15)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=gh,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, 8632039);