#8110558

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4624 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8017754);