#8903074

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

4646 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8810270);