#8978374

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

4400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8885570);