#8343561

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	#38024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4641 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=gi,f^2=gi,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, 8250757);