#8342253

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

5551 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gh,e^2=hi,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, 8249449);