#8257970

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=hi,f^2=ghi,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, 8165166);