#8416781

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8323977);