#8718913

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	#58246	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(3,10,5,8)(4,12,13,9)(11,15,14,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,34,38,42)(35,43,46,39)(36,45,47,40)(37,41,48,44);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,30)(28,29)(31,32)(33,35,38,46)(34,39,42,43)(36,44,47,41)(37,45,48,40);(4,13)(9,12)(11,14)(15,16)(18,23)(19,24)(25,32)(27,29)(33,35,38,46)(34,39,42,43)(36,44,47,41)(37,45,48,40);(1,4)(2,9)(3,11)(5,14)(6,12)(7,13)(8,15)(10,16)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28)(33,35)(34,39)(36,44)(37,40)(38,46)(41,47)(42,43)(45,48);(1,5,7,3)(2,8,6,10)(4,14,13,11)(9,15,12,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,36,38,47)(34,40,42,45)(35,44,46,41)(37,43,48,39);(1,6,7,2)(3,10,5,8)(4,9,13,12)(11,15,14,16)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)(33,37)(34,41)(35,45)(36,39)(38,48)(40,46)(42,44)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8626109);