#8131533

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

4782 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8038729);