#10325327

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	#56404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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)(3,8)(4,7)(5,6)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,31,34)(27,32,38,33)(28,39,29,30)(35,37,36,40);(1,3,6,4)(2,7,5,8)(11,19)(13,23)(15,24)(17,20)(25,27,31,38)(26,32,34,33)(28,36,29,35)(30,37,39,40);(1,2,6,5)(3,7,4,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,28,31,29)(26,30,34,39)(27,35,38,36)(32,40,33,37);(1,2,6,5)(3,8,4,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,29,31,28)(26,30,34,39)(27,36,38,35)(32,40,33,37);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,30)(26,28)(27,37)(29,34)(31,39)(32,36)(33,35)(38,40);(1,5,6,2)(3,7,4,8)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,38)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=g,f^2=g,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, 10232523);