#10482454

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	#55126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389650);