#8700473

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

4854 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,15,13,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,19,22,27)(18,23,20,25)(21,26,30,29)(24,31,28,32);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,12,15)(17,18,22,20)(19,23,27,25)(21,24,30,28)(26,31,29,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,15,12,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32);(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607669);