#8180510

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

3986 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=h,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, 8087706);