#10324094

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	#37729	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=gh,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, 10231290);