#10195449

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	#41927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,35,40,48)(34,41,44,45)(36,46,37,39)(38,47,42,43);(2,9)(4,15)(8,12)(11,13)(18,20)(19,27)(25,30)(26,29)(33,36,40,37)(34,42,44,38)(35,46,48,39)(41,43,45,47);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,37,40,36)(34,42,44,38)(35,39,48,46)(41,43,45,47);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,38,40,42)(34,37,44,36)(35,47,48,43)(39,45,46,41);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,39)(34,43)(35,36)(37,48)(38,41)(40,46)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10102645);