#8607434

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	#53558	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8514630);