#8323144

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

4858 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)(3,8)(4,6)(5,7)(10,13)(11,12)(17,18,23,25)(19,26,29,22)(20,21,24,32)(27,30,31,28);(1,2,6,4)(3,8,5,7)(9,10,14,13)(11,16,12,15)(17,19)(18,22)(20,27)(21,28)(23,29)(24,31)(25,26)(30,32);(1,2,6,4)(3,7,5,8)(9,11,14,12)(10,15,13,16)(17,20,23,24)(18,21,25,32)(19,27,29,31)(22,28,26,30);(1,3,6,5)(2,7,4,8)(9,12,14,11)(10,16,13,15)(17,18)(19,22)(20,21)(23,25)(24,32)(26,29)(27,28)(30,31);(1,4,6,2)(3,7,5,8)(9,13,14,10)(11,16,12,15)(17,21)(18,24)(19,28)(20,25)(22,31)(23,32)(26,27)(29,30);(1,5,6,3)(2,7,4,8)(17,22)(18,19)(20,30)(21,31)(23,26)(24,28)(25,29)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8230340);