#8202434

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×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109630);