#8633781

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	#47505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(33,34,39,41)(35,40,44,36)(37,46,45,42)(38,43,47,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35)(34,40)(36,41)(37,47)(38,45)(39,44)(42,43)(46,48);(2,5)(8,10)(9,12)(15,16)(18,23)(20,29)(24,31)(26,32)(34,41)(35,44)(37,45)(43,48);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36)(34,35)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,37,39,45)(34,42,41,46)(35,38,44,47)(36,43,40,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,38)(34,43)(35,45)(36,46)(37,44)(39,47)(40,42)(41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=h,f^2=hi,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, 8540977);