#8997247

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

5386 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8904443);