#8393527

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300723);