#8923680

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

4598 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8830876);