#8095314

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

5595 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)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,28)(20,30)(21,24)(22,29)(23,25)(26,31)(27,32);(2,4)(6,7)(9,11,12,16)(10,13,15,14)(17,19,23,21)(18,24,25,28)(20,27,31,29)(22,26,32,30);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,16)(13,15)(18,25)(19,21)(26,30)(27,29);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,20,23,31)(18,26,25,30)(19,29,21,27)(22,28,32,24);(17,21,23,19)(18,24,25,28)(20,27,31,29)(22,30,32,26);(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,31)(25,29)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002510);