#7763432

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

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, 7670628);