#8599216

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	#40536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4869 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)(3,4,8,6)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19)(18,24)(20,30)(21,31)(22,25)(23,27)(26,28)(29,32);(2,7)(4,6)(9,10,12,13)(11,14,15,16)(18,25)(19,27)(21,32)(28,30);(2,7)(3,8)(11,15)(14,16)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,31,24,29);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,30)(27,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,22)(18,27)(19,25)(20,29)(21,28)(23,24)(26,31)(30,32)

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,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, 8506412);