#8139528

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4799 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)(6,8)(9,10,15,14)(11,18,21,16)(12,17,13,22)(19,24,20,23)(27,34)(29,37)(31,32)(33,36);(3,7)(6,8)(9,11)(10,16)(12,19)(13,20)(14,18)(15,21)(17,23)(22,24)(25,26)(27,32)(28,30)(29,33)(31,34)(35,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,14,22)(11,19,21,20)(16,23,18,24)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,17,14,22)(11,20,21,19)(16,23,18,24)(25,28,30,26)(27,32,34,31)(29,36,37,33)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,14,15,10)(11,18,21,16)(12,22,13,17)(19,24,20,23)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(9,11)(10,16)(12,19)(13,20)(14,18)(15,21)(17,23)(22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

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

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, 8046724);