#8152236

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	#36101	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4875 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=g,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^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, 8059432);