#8621670

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	#46080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,e^2=ghi,f^2=g,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, 8528866);