#7658121

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,e^2=g,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, 7565317);