#7651719

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4401 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)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,33)(27,34,37,30)(28,38,29,32)(35,40,36,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,27)(26,30)(28,35)(29,36)(31,37)(32,40)(33,34)(38,39);(11,20)(13,21)(15,19)(17,24)(25,28,31,29)(26,32,33,38)(27,35,37,36)(30,40,34,39);(9,12)(10,16)(11,21)(13,20)(14,23)(15,17)(18,22)(19,24)(25,26)(27,30)(28,32)(29,38)(31,33)(34,37)(35,40)(36,39);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,29,31,28)(26,32,33,38)(27,36,37,35)(30,40,34,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,12,20,23)(15,22,19,16)(25,30)(26,27)(28,39)(29,40)(31,34)(32,36)(33,37)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558915);