#8595758

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

5064 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)(5,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32);(1,2)(3,8)(4,6)(5,7)(10,11)(14,15)(17,19,22,26)(18,23,25,30)(20,27,24,31)(21,28,29,32);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,14,16,15)(19,26)(23,30)(27,31)(28,32);(1,2,4,6)(3,5,7,8)(9,11,13,10)(12,15,16,14)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=g,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, 8502954);