#7712678

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×#540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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)(10,16)(14,24)(17,20)(18,22)(26,32)(27,30)(29,37)(36,40);(1,2,5,6)(3,7,4,8)(9,10,15,16)(11,20,12,17)(13,18,23,22)(14,19,24,21)(25,26)(27,28)(29,34)(30,33)(31,32)(35,36)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,17,16,20)(13,19,23,21)(14,22,24,18)(25,27,31,30)(26,33,32,28)(29,38,37,35)(34,36,39,40);(1,4,5,3)(2,8,6,7)(9,12,15,11)(10,17,16,20)(13,19,23,21)(14,18,24,22)(25,28,31,33)(26,27,32,30)(29,36,37,40)(34,35,39,38);(9,13,15,23)(10,18,16,22)(11,21,12,19)(14,17,24,20)(25,29,31,37)(26,34,32,39)(27,35,30,38)(28,36,33,40);(9,14,15,24)(10,19,16,21)(11,22,12,18)(13,17,23,20)(25,30,31,27)(26,33,32,28)(29,38,37,35)(34,40,39,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619874);