#8750807

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=gi,e^2=h,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, 8658003);