#8115477

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

4763 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022673);