#8135382

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

5648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8042578);