#8898896

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8806092);