#8188198

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(34,38)(35,41)(40,46)(42,43);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(2,8)(3,9)(10,13)(11,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35,37,41)(34,39,38,45)(36,43,44,42)(40,48,46,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,36,37,44)(34,40,38,46)(35,42,41,43)(39,47,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095394);