#8662938

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	#55165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8570134);