#8445989

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	#59141	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=gi,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, 8353185);