#8751592

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

4625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658788);