#7915874

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^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, 7823070);