#8093662

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000858);