#8132461

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

4893 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)(9,10)(11,13)(12,15)(14,16)(17,18,23,21)(19,25,29,24)(20,30,26,22)(27,28,31,32);(1,2)(3,6)(4,5)(7,8)(10,11)(14,15)(17,19)(18,24)(20,27)(21,25)(22,28)(23,29)(26,31)(30,32);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(17,19)(18,25)(20,31)(21,24)(22,28)(23,29)(26,27)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20)(18,22)(19,27)(21,30)(23,26)(24,28)(25,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,11,15)(17,21,23,18)(19,25,29,24)(20,30,26,22)(27,32,31,28);(1,3,4,7)(2,6,5,8)(17,22,23,30)(18,26,21,20)(19,28,29,32)(24,31,25,27)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=g,f^2=gi,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, 8039657);