#7712719

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4408 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)(9,10,15,12)(11,17,13,16)(14,20,22,18)(19,23,21,24)(28,36)(32,37)(34,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(26,30)(31,33)(32,37)(38,39);(1,4,5,2)(3,7,8,6)(10,12)(11,13)(18,20)(19,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,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, 7619915);