#7987058

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	#34715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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)(9,10,15,17)(11,16,12,14)(13,21,22,18)(19,24,20,23)(25,26,32,30)(27,35,29,33)(28,34,31,36)(37,40,38,39);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,21,24)(25,27,32,29)(26,33,30,35)(28,38,31,37)(34,40,36,39);(10,17)(11,12)(13,22)(23,24)(25,28,32,31)(26,34,30,36)(27,37,29,38)(33,39,35,40);(9,12,15,11)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,29,32,27)(26,35,30,33)(28,38,31,37)(34,40,36,39);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,19)(12,20)(14,24)(15,22)(16,23)(17,21)(25,30,32,26)(27,35,29,33)(28,34,31,36)(37,39,38,40);(9,14,15,16)(10,11,17,12)(13,23,22,24)(18,20,21,19)(25,31,32,28)(26,36,30,34)(27,38,29,37)(33,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894254);