#8772532

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

3848 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679728);