#7642117

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	#37055	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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)(11,17)(12,16)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26,31,35)(27,34,30,32)(28,38,33,29)(36,40,39,37);(1,3,5,8)(2,6,4,7)(12,20)(16,22)(19,24)(21,23)(25,27,31,30)(26,32,35,34)(28,39,33,36)(29,37,38,40);(10,15)(11,18)(16,22)(19,24)(25,28,31,33)(26,29,35,38)(27,36,30,39)(32,40,34,37);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,24)(22,23)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,36)(34,39);(9,12,14,20)(10,16,15,22)(11,19,18,24)(13,21,17,23)(25,27,31,30)(26,34,35,32)(28,36,33,39)(29,37,38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,30,31,27)(26,32,35,34)(28,39,33,36)(29,40,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

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, 7549313);