#8105971

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	#43932	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5007 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013167);