#8873743

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=i,e^2=gi,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, 8780939);