#8231469

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	#53004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gi,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, 8138665);