#7895145

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

4445 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7802341);