#7927474

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	#44253	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(26,32)(28,34)(33,35)(36,38);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,26,31,32)(27,35,29,33)(28,30,34,40)(36,39,38,37);(1,2,5,4)(3,6,8,7)(10,16)(15,18)(17,20)(23,24)(25,27,31,29)(26,33,32,35)(28,36,34,38)(30,37,40,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28,31,34)(26,30,32,40)(27,36,29,38)(33,37,35,39);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,17,16,20)(11,19,13,21)(15,23,18,24)(25,29,31,27)(26,33,32,35)(28,38,34,36)(30,37,40,39);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,30)(26,34)(27,37)(28,32)(29,39)(31,40)(33,38)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834670);