#8131221

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	#53526	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,5)(4,12)(6,7)(8,13)(9,10)(11,16)(14,15)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(34,38)(39,44)(40,45)(46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,14)(12,16)(13,15)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(35,41)(39,44)(42,47)(46,48);(2,7)(3,9)(4,13)(14,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,13)(14,15)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,4,6,13)(2,8,7,12)(3,11,9,15)(5,14,10,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5,6,10)(2,9,7,3)(4,14,13,16)(8,15,12,11)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038417);