#9862952

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

4510 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770148);