#8746034

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	#52543	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653230);