#8970015

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

4325 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)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38);(1,2)(3,4)(5,7)(6,8)(10,15)(12,23)(16,19)(18,21)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(1,3)(2,6)(4,7)(5,8)(10,15)(11,20)(12,23)(13,24)(26,27)(28,35)(29,37)(38,39);(3,8)(4,6)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,2)(3,4)(5,7)(6,8)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8877211);