#8594954

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	#42296	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8502150);