#8605592

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

4859 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)(11,15)(13,16)(17,18,24,28)(19,21,22,25)(20,29,30,26)(23,27,32,31);(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,19)(18,25)(20,27)(21,28)(22,24)(23,29)(26,32)(30,31);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,20,24,30)(18,26,28,29)(19,27,22,31)(21,23,25,32);(2,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,19)(20,23)(22,28)(24,25)(26,27)(29,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,22,24,19)(18,21,28,25)(20,27,30,31)(23,26,32,29);(1,2)(3,5)(4,6)(7,8)(17,23)(18,27)(19,29)(20,25)(21,30)(22,26)(24,32)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,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, 8512788);