#8634550

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	#39191	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,e^2=h,f^2=gi,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, 8541746);