#8445855

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353051);