#8898548

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,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, 8805744);