#7931612

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

4243 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)(11,16)(12,21)(13,22)(14,18)(15,17)(19,23)(20,24)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,19)(10,16,17,23)(12,20,22,14)(13,18,21,24);(1,3,5,4)(2,7,6,8)(10,17)(11,19)(12,22)(18,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,20)(14,19)(15,22)(16,18)(17,21)(23,24)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,18)(14,23)(15,21)(16,20)(17,22)(19,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,3)(2,8,6,7)(9,14,15,20)(10,18,17,24)(11,12,19,22)(13,16,21,23)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=ghi,e^2=g,f^2=gh,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, 7838808);