#10069035

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

4699 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)(3,7)(4,6)(5,8)(11,17)(15,18)(19,23)(22,24)(25,26,31,32)(27,34,30,29)(28,33,37,39)(35,40,36,38);(1,3,5,4)(2,6,8,7)(9,10,14,12)(11,18,17,15)(13,16,21,20)(19,24,23,22)(25,27,31,30)(26,29,32,34)(28,36,37,35)(33,40,39,38);(1,3,5,4)(2,7,8,6)(10,12)(15,18)(16,20)(22,24)(26,32)(28,37)(29,34)(35,36);(1,2)(3,7)(4,6)(5,8)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(1,2)(3,6)(4,7)(5,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29)(26,27)(28,38)(30,32)(31,34)(33,35)(36,39)(37,40);(1,4,5,3)(2,6,8,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,27)(26,34,32,29)(28,36,37,35)(33,40,39,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9976231);