#8968128

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	#57961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8875324);