#9053859

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	#51581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,e^2=i,f^2=g,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, 8961055);