#7972741

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
prime radical	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7879937);