#8358120

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

5108 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)(14,16)(17,18,24,28)(19,22,23,25)(20,27,21,26)(29,32,31,30);(1,2,5,6)(3,7,8,4)(11,15)(14,16)(17,19,24,23)(18,25,28,22)(20,29,21,31)(26,32,27,30);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,28,27)(19,29,23,31)(22,30,25,32);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,28,26)(19,29,23,31)(22,30,25,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,22,24,25)(18,19,28,23)(20,30,21,32)(26,29,27,31);(1,4)(2,3)(5,7)(6,8)(17,23,24,19)(18,25,28,22)(20,29,21,31)(26,30,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=hi,e^2=g,f^2=g,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, 8265316);