#7744638

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	#35567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(9,10,14,13)(11,15,18,16)(12,17,20,21)(19,22,24,23)(25,26,31,33)(27,32,29,30)(28,36,39,34)(35,38,37,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(25,27,31,29)(26,32,33,30)(28,35,39,37)(34,40,36,38);(1,3,5,4)(2,7,6,8)(11,18)(12,20)(15,16)(17,21)(26,33)(27,29)(34,36)(35,37);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,28)(26,34)(27,35)(29,37)(30,38)(31,39)(32,40)(33,36);(9,12,14,20)(10,17,13,21)(11,19,18,24)(15,22,16,23)(25,29,31,27)(26,32,33,30)(28,37,39,35)(34,40,36,38);(9,13,14,10)(11,15,18,16)(12,21,20,17)(19,22,24,23)(25,30,31,32)(26,27,33,29)(28,38,39,40)(34,35,36,37)

Polycyclic presentation:

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

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, 7651834);