#8459147

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

5040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=hi,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, 8366343);