#10323160

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	#51524	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26,31,29)(27,32,38,37)(28,34,39,33)(30,40,36,35);(1,2)(3,7)(4,5)(6,8)(9,11,13,19)(10,14,15,17)(12,21,22,18)(16,24,20,23)(25,27,31,38)(26,32,29,37)(28,36,39,30)(33,40,34,35);(2,5)(6,7)(10,15)(14,17)(16,20)(23,24)(25,28)(26,33)(27,36)(29,34)(30,38)(31,39)(32,40)(35,37);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,29,31,26)(27,37,38,32)(28,34,39,33)(30,40,36,35);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,28)(26,34)(27,36)(29,33)(30,38)(31,39)(32,35)(37,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,22)(10,16,15,20)(11,18,19,21)(14,23,17,24)(25,30,31,36)(26,35,29,40)(27,28,38,39)(32,33,37,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,f^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, 10230356);