#8388314

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

4909 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)(17,18)(19,27)(20,28)(21,26)(22,24)(23,29)(25,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(17,19,22,29)(18,23,24,27)(20,26,31,30)(21,28,32,25);(4,8)(6,7)(10,14)(12,15)(18,24)(21,32)(23,27)(25,28);(2,3)(6,7)(10,14)(11,16)(17,20)(18,25)(19,26)(21,23)(22,31)(24,28)(27,32)(29,30);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,32)(26,31);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,15)(14,16)(17,21,22,32)(18,26,24,30)(19,28,29,25)(20,23,31,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295510);