#8095876

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

5173 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:

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

Polycyclic presentation:

2^9|d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003072);