#8272698

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

4882 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8179894);