#9057896

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	#55806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4249 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)(4,8)(5,6)(9,10,14,17)(11,12,15,18)(13,16,22,21)(19,20,23,24)(26,31)(29,37)(32,38)(36,40);(1,2,5,6)(3,7,8,4)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(28,35)(32,38)(34,39)(36,40);(1,3)(2,4)(5,8)(6,7)(9,10,14,17)(11,18,15,12)(13,21,22,16)(19,20,23,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965092);