#8227312

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	#55589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,15,16,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32);(2,7)(3,12)(4,10)(14,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(2,7)(5,9)(8,11)(14,15)(18,22)(23,25)(24,28)(30,31)(34,39)(37,46)(40,43)(44,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,15)(10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,15,11,14)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,16,11,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

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

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, 8134508);