#8131522

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
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=ghi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038718);