#10519360

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	#46099	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=gh,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, 10426556);