#8254700

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=h,e^2=hi,f^2=gi,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, 8161896);