#8969768

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	#45892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8876964);