#8922532

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	#46231	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829728);