#8075446

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

4470 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982642);