#8630902

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	#50154	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=g,e^2=hi,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, 8538098);