#8093290

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
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4764 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000486);