#10216615

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	#49801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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)(3,7)(4,6)(5,8)(10,14)(15,17)(16,22)(23,24)(25,26,30,28)(27,31,35,32)(29,36,37,33)(34,39,40,38);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(25,27,30,35)(26,32,28,31)(29,34,37,40)(33,39,36,38);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,28,30,26)(27,31,35,32)(29,33,37,36)(34,39,40,38);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=gi,f^2=gi,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, 10123811);