#7642115

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

4427 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)(11,17)(12,16)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26,32,34)(27,35,29,30)(28,33,31,40)(36,39,37,38);(1,3,5,8)(2,6,4,7)(12,20)(16,22)(19,24)(21,23)(25,27,32,29)(26,30,34,35)(28,37,31,36)(33,39,40,38);(10,15)(11,18)(16,22)(19,24)(25,28,32,31)(26,33,34,40)(27,36,29,37)(30,38,35,39);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,24)(22,23)(25,29,32,27)(26,30,34,35)(28,36,31,37)(33,39,40,38);(9,12,14,20)(10,16,15,22)(11,19,18,24)(13,21,17,23)(25,30)(26,27)(28,38)(29,34)(31,39)(32,35)(33,36)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,31,32,28)(26,33,34,40)(27,36,29,37)(30,39,35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549311);