#8339875

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

3905 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247071);