#7737816

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 7645012);