#10480817

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	#50099	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 10388013);