#7654108

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	#39883	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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)(10,12)(11,16)(17,20)(19,24)(25,26,31,35)(27,36,38,32)(28,29,33,39)(30,34,40,37);(1,2,5,6)(3,7,4,8)(9,10,15,12)(11,13,16,18)(14,17,23,20)(19,21,24,22)(25,27,31,38)(26,32,35,36)(28,30,33,40)(29,37,39,34);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,26,31,35)(27,36,38,32)(28,29,33,39)(30,34,40,37);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,28,31,33)(26,29,35,39)(27,30,38,40)(32,37,36,34);(1,3,5,4)(2,7,6,8)(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29)(26,33)(27,37)(28,35)(30,36)(31,39)(32,40)(34,38);(1,4,5,3)(2,8,6,7)(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)(25,30)(26,34)(27,33)(28,38)(29,32)(31,40)(35,37)(36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561304);