#7645849

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	#41022	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553045);