#8898890

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	#57248	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,15)(13,16)(17,18,23,24)(19,26,20,25)(21,27,31,29)(22,30,32,28)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(1,3)(2,8)(4,13)(5,6)(7,12)(9,15)(10,16)(11,14)(18,24)(21,31)(25,26)(28,30)(34,40)(35,45)(37,43)(42,47);(2,9)(3,12)(6,10)(11,13)(17,19,23,20)(18,25,24,26)(21,28,31,30)(22,27,32,29)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,4)(2,10)(3,13)(5,15)(6,9)(7,14)(8,16)(11,12)(17,20,23,19)(18,26,24,25)(21,28,31,30)(22,27,32,29)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,5)(2,11)(3,10)(4,8)(6,12)(7,16)(9,13)(14,15)(17,21,23,31)(18,27,24,29)(19,28,20,30)(22,25,32,26)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,6,7,10)(2,4,9,14)(3,5,12,16)(8,13,15,11)(17,22)(18,28)(19,29)(20,27)(21,25)(23,32)(24,30)(26,31)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806086);