#8375594

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

4907 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8282790);