#8455760

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	#38509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4559 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)(9,10,15,19)(11,14,18,23)(12,22,21,16)(13,17,20,24)(25,26,31,35)(27,36,39,32)(28,30,34,40)(29,38,37,33);(3,7)(6,8)(9,11,15,18)(10,14,19,23)(12,13,21,20)(16,24,22,17)(25,27,31,39)(26,32,35,36)(28,37,34,29)(30,38,40,33);(1,2)(3,6)(4,5)(7,8)(9,12,15,21)(10,16,19,22)(11,20,18,13)(14,17,23,24)(25,28,31,34)(26,30,35,40)(27,37,39,29)(32,38,36,33);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,20)(13,21)(15,18)(16,17)(19,23)(22,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,15,20)(10,17,19,24)(11,21,18,12)(14,16,23,22)(25,29)(26,33)(27,28)(30,32)(31,37)(34,39)(35,38)(36,40);(9,14)(10,18)(11,19)(12,17)(13,16)(15,23)(20,22)(21,24)(25,30)(26,34)(27,38)(28,35)(29,32)(31,40)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8362956);