#8724880

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
prime radical	#35995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8632076);