#8630903

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	#53834	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)(3,14)(4,11)(5,10)(6,15)(7,9)(8,13)(12,16)(17,18,22,25)(19,21,24,31)(20,27,30,23)(26,32,28,29)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(1,3,8,7)(2,9,13,14)(4,15,5,12)(6,11,16,10)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(34,40)(35,45)(36,42)(43,47);(1,4,8,5)(2,10,13,11)(3,15,7,12)(6,14,16,9)(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,5,8,4)(2,11,13,10)(3,15,7,12)(6,14,16,9)(17,19)(18,21)(20,26)(22,24)(23,29)(25,31)(27,32)(28,30)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,6,8,16)(2,12,13,15)(3,10,7,11)(4,9,5,14)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,7,8,3)(2,9,13,14)(4,12,5,15)(6,11,16,10)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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, 8538099);