#10144532

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	#49743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10051728);