#8730827

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

4934 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)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(18,22)(19,25)(24,27)(26,28);(1,3)(2,6)(4,7)(5,8)(10,13)(14,15)(18,22)(23,30)(24,27)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,25,30)(20,24,29,27)(26,31,28,32);(3,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31);(1,3)(2,7)(4,6)(5,8)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8638023);