#8375640

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,27)(19,24,28,31)(20,21,25,32)(22,26,29,30)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(2,7)(3,11)(4,14)(13,16)(17,19,23,28)(18,24,27,31)(20,29,25,22)(21,30,32,26)(34,39)(40,43)(41,46)(47,48);(2,7)(5,9)(8,10)(13,16)(19,28)(22,29)(24,31)(26,30)(33,35,38,36)(34,40,39,43)(37,44,45,42)(41,48,46,47);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20)(18,21)(19,29)(22,28)(23,25)(24,30)(26,31)(27,32)(33,36,38,35)(34,40,39,43)(37,44,45,42)(41,47,46,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21)(18,25)(19,26)(20,27)(22,31)(23,32)(24,29)(28,30)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22,23,29)(18,26,27,30)(19,20,28,25)(21,31,32,24)(33,37,38,45)(34,41,39,46)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8282836);