#10038477

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
prime radical	#41950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4355 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)(4,5)(6,8)(11,17)(15,18)(19,23)(22,24)(25,26,30,28)(27,32,35,31)(29,36,38,33)(34,39,37,40);(2,5)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,35)(26,31,28,32)(29,34,38,37)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,18)(16,20)(22,24)(25,27,30,35)(26,31,28,32)(29,34,38,37)(33,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28,30,26)(27,31,35,32)(29,36,38,33)(34,39,37,40);(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,38)(26,33,28,36)(27,34,35,37)(31,39,32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,f^2=h,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, 9945673);