#8458502

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4670 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)(9,10,16,15)(11,20,22,17)(12,18,13,23)(14,24,21,19)(25,26)(27,32)(28,39)(29,34)(30,35)(31,36)(33,38)(37,40);(2,3)(6,7)(9,11)(10,17)(12,14)(13,21)(15,20)(16,22)(18,19)(23,24)(25,27,31,35)(26,32,36,30)(28,29,38,37)(33,40,39,34);(1,3,5,2)(4,7,8,6)(9,12,16,13)(10,18,15,23)(11,14,22,21)(17,19,20,24)(25,28,31,38)(26,33,36,39)(27,37,35,29)(30,40,32,34);(1,4,5,8)(2,6,3,7)(9,13,16,12)(10,18,15,23)(11,21,22,14)(17,19,20,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,34)(39,40);(9,14,16,21)(10,19,15,24)(11,12,22,13)(17,18,20,23)(25,29,31,37)(26,34,36,40)(27,38,35,28)(30,39,32,33);(9,15,16,10)(11,20,22,17)(12,23,13,18)(14,24,21,19)(25,30,31,32)(26,35,36,27)(28,40,38,34)(29,39,37,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gi,f^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, 8365698);