#7954508

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	C₂×#562	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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)(11,19)(12,22)(13,24)(14,18)(15,21)(16,23)(17,20)(25,26)(27,32)(28,29)(30,33)(31,35)(34,40)(36,37)(38,39);(1,2,5,6)(3,7,4,8)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,17,22,24)(25,27,31,30)(26,32,35,33)(28,36,34,38)(29,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,14,23)(10,16,18,22)(11,13,21,20)(15,24,19,17)(25,27,31,30)(26,33,35,32)(28,38,34,36)(29,37,40,39);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,20)(13,21)(14,23)(15,17)(18,22)(19,24)(25,28,31,34)(26,29,35,40)(27,36,30,38)(32,37,33,39);(1,3,5,4)(2,7,6,8)(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,22,19,16)(25,29)(26,34)(27,37)(28,35)(30,39)(31,40)(32,38)(33,36);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,24)(14,18)(17,20)(19,21)(22,23)(25,30,31,27)(26,33,35,32)(28,38,34,36)(29,39,40,37)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gh,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, 7861704);