#8119926

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

5143 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:

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

Polycyclic presentation:

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

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, 8027122);