#8095821

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

4897 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=i,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, 8003017);