#7656936

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	#42952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4443 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564132);