#7781044

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	#46224	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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)(9,10,15,19)(11,16,14,18)(12,13,17,24)(20,21,22,23)(25,26,32,29)(27,31,35,37)(28,34,30,33)(36,39,38,40);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,19,18)(12,20,17,22)(13,21,24,23)(25,27)(26,31)(28,38)(29,37)(30,36)(32,35)(33,39)(34,40);(11,14)(16,18)(20,22)(21,23)(25,28,32,30)(26,33,29,34)(27,36,35,38)(31,40,37,39);(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,29,32,26)(27,37,35,31)(28,33,30,34)(36,40,38,39);(1,2,5,4)(3,6,8,7)(9,13)(10,17)(11,21)(12,19)(14,23)(15,24)(16,22)(18,20)(25,30,32,28)(26,34,29,33)(27,36,35,38)(31,40,37,39);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,23,24,21)(25,31,32,37)(26,35,29,27)(28,39,30,40)(33,36,34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7688240);