#8258110

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4641 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8165306);