#8932988

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4507 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840184);