#8712756

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

(1,2,6,7)(3,9,10,5)(4,12,14,8)(11,15,16,13)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,41)(36,39)(37,42)(38,44)(40,45)(43,47)(46,48);(2,7)(3,10)(8,12)(11,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,38)(36,43)(37,44)(39,46)(40,47)(41,42)(45,48);(2,7)(3,10)(8,12)(11,16)(19,28)(20,29)(24,30)(25,32)(33,36,37,45)(34,39,42,40)(35,43,44,48)(38,46,41,47);(1,3,6,10)(2,5,7,9)(4,11,14,16)(8,15,12,13)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36)(34,40)(35,43)(37,45)(38,47)(39,42)(41,46)(44,48);(1,4,6,14)(2,8,7,12)(3,11,10,16)(5,15,9,13)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,35)(34,41)(36,43)(37,44)(38,42)(39,47)(40,46)(45,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,16)(11,12)(14,15)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(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=i,c^2=gh,d^2=h,e^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, 8619952);