#8873759

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3824 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780955);