#8726724

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	#52294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gi,e^2=i,f^2=gi,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, 8633920);