#7732038

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

4193 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)(9,10)(11,21)(12,15)(13,18)(14,19)(16,20)(17,23)(22,24)(26,27)(29,37)(31,33)(35,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,17)(12,14)(13,22)(15,24)(16,23)(18,19)(20,21)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3,5,4)(2,7,6,8)(9,12,16,13)(10,15,20,18)(11,22,23,14)(17,19,21,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,4,5,3)(2,8,6,7)(9,13,16,12)(10,18,20,15)(11,22,23,14)(17,19,21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,14,16,22)(10,19,20,24)(11,12,23,13)(15,17,18,21)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40);(9,15,16,18)(10,12,20,13)(11,19,23,24)(14,17,22,21)

Polycyclic presentation:

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

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, 7639234);