#7737808

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

4344 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)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,3,5,8)(2,6,4,7)(27,34)(28,36)(30,33)(31,38);(1,4,5,2)(3,7,8,6)(10,15)(11,18)(17,20)(19,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,38)(37,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7645004);