#8346436

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	#35703	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4797 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8253632);