#8811196

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	#48247	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8718392);