#10480866

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	#57209	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,f^2=g,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, 10388062);