#7712136

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	#36404	C₂
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619332);