#8622152

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8529348);