#7657574

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

4197 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)(9,10,15,13)(11,18,16,12)(14,17,22,21)(19,24,23,20)(25,26)(27,30)(28,36)(29,33)(31,34)(32,39)(35,37)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(1,2,5,4)(3,6,8,7)(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(1,2,5,4)(3,6,8,7)(9,13,15,10)(11,18,16,12)(14,17,22,21)(19,20,23,24)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=i,e^2=gi,f^2=gi,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, 7564770);