#8818077

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

4566 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)(9,10,14,13)(11,15,19,18)(12,20,22,16)(17,24,21,23)(25,26,32,38)(27,35,29,33)(28,37,31,34)(30,36,40,39);(2,6)(3,7)(10,13)(15,18)(16,20)(23,24)(25,27,32,29)(26,33,38,35)(28,39,31,36)(30,37,40,34);(1,3)(2,4)(5,7)(6,8)(9,11,14,19)(10,15,13,18)(12,21,22,17)(16,24,20,23)(25,28,32,31)(26,34,38,37)(27,39,29,36)(30,35,40,33);(1,4)(2,3)(5,8)(6,7)(9,12,14,22)(10,16,13,20)(11,17,19,21)(15,23,18,24)(25,29,32,27)(26,35,38,33)(28,39,31,36)(30,37,40,34);(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,30,32,40)(26,36,38,39)(27,34,29,37)(28,33,31,35);(9,13,14,10)(11,18,19,15)(12,20,22,16)(17,24,21,23)(25,31,32,28)(26,37,38,34)(27,39,29,36)(30,35,40,33)

Polycyclic presentation:

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

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, 8725273);