#8902530

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

3850 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809726);