#8445837

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

4303 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=i,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, 8353033);