#8739189

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=h,f^2=gh,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, 8646385);