#8924359

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
prime radical	#58135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=gi,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, 8831555);