#8751370

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658566);