#7783228

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	#45373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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)(11,19)(12,16)(13,14)(15,17)(18,24)(20,23)(21,22)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,19)(13,20,24,22)(14,21,18,23)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(10,17)(11,12)(14,18)(20,22)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,39,38,37);(9,12,15,11)(10,16,17,19)(13,20,24,22)(14,23,18,21)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,20)(12,22)(14,17)(15,24)(16,23)(19,21)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(9,14,15,18)(10,13,17,24)(11,21,12,23)(16,20,19,22)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=i,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, 7690424);