#8932825

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
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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=h,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,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, 8840021);