#8600874

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

4839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8508070);