#8893483

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

4079 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)(3,5,11,7)(4,9,14,16)(10,13,12,15)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,39,38)(35,41,45,40)(36,46,44,42)(37,43,48,47);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,39,45)(34,40,38,41)(36,37,44,48)(42,43,46,47);(2,8)(5,7)(9,16)(13,15)(19,26)(23,30)(27,31)(28,32)(33,35,39,45)(34,41,38,40)(36,37,44,48)(42,47,46,43);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36)(34,42)(35,37)(38,46)(39,44)(40,43)(41,47)(45,48);(1,4)(2,9)(3,10)(5,13)(6,14)(7,15)(8,16)(11,12)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,43)(35,44)(36,45)(38,47)(39,48)(40,46)(41,42);(1,5,6,7)(2,3,8,11)(4,13,14,15)(9,10,16,12)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38,39,34)(35,41,45,40)(36,46,44,42)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,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, 8800679);