#8718973

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=i,e^2=g,f^2=h,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, 8626169);