#7642114

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	#37228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

2^9|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, 7549310);