#8718129

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	#46402	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=i,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, 8625325);