#10323949

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	#42111	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=gh,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, 10231145);