#7927703

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=i,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, 7834899);