#8924395

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

4465 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831591);