#8325947

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

4915 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=h,e^2=h,f^2=hi,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, 8233143);