#8125545

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

5069 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)(17,18,23,27)(19,28,30,24)(20,21,31,25)(22,32,29,26);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19)(18,24)(20,22)(21,26)(23,30)(25,32)(27,28)(29,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,31)(18,25,27,21)(19,22,30,29)(24,32,28,26);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,25)(19,29)(21,27)(22,30)(23,31)(24,26)(28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,25)(18,20,27,31)(19,26,30,32)(22,28,29,24);(1,3)(2,6)(4,7)(5,8)(17,22,23,29)(18,26,27,32)(19,20,30,31)(21,28,25,24)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^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,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, 8032741);