#10349013

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

4573 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10256209);