#8417102

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3769 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324298);