#8475189

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

5248 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8382385);