#10096932

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

4390 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10004128);