#7658925

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566121);