#8165258

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

5032 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8072454);