#8922559

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

4573 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=i,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, 8829755);