#7656565

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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)(12,20)(15,22)(18,24)(21,23)(25,26,31,30)(27,35,38,32)(28,33,37,39)(29,34,36,40);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,34)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,34,35,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,33,35,39);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,34,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=h,f^2=h,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, 7563761);