#8922579

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=i,e^2=gi,f^2=h,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, 8829775);