#10368674

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

4107 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275870);