#8115438

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	#42982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022634);