#10143700

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	#44209	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=ghi,e^2=gh,f^2=h,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, 10050896);