#7658619

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	#42454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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)(11,20)(12,18)(13,16)(14,19)(15,17)(21,24)(22,23)(25,26,30,34)(27,35,36,31)(28,32,29,33)(37,40,38,39);(1,2,5,6)(3,7,4,8)(9,11,16,15)(10,17,13,20)(12,22,14,21)(18,24,19,23)(27,36)(31,35)(37,38)(39,40);(1,3,5,4)(2,7,6,8)(9,12,16,14)(10,18,13,19)(11,21,15,22)(17,23,20,24)(25,27,30,36)(26,31,34,35)(28,37,29,38)(32,39,33,40);(1,4,5,3)(2,8,6,7)(9,13,16,10)(11,17,15,20)(12,19,14,18)(21,23,22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(9,14,16,12)(10,19,13,18)(11,21,15,22)(17,23,20,24)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,39,35,40);(9,15,16,11)(10,17,13,20)(12,22,14,21)(18,23,19,24)(25,29,30,28)(26,33,34,32)(27,38,36,37)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=i,f^2=i,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, 7565815);