#10480864

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	#50095	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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)(19,20)(21,29)(22,23)(24,27)(25,31)(26,32)(28,30)(33,34,38,41)(35,36,43,39)(37,45,47,40)(42,48,46,44);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(18,23)(20,24)(25,29)(28,32)(33,34,38,41)(35,36,43,39)(37,40,47,45)(42,44,46,48);(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,48)(44,45)(46,47);(1,4,7,2)(3,8,6,10)(5,12,13,9)(11,15,14,16)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,36,38,39)(34,35,41,43)(37,44,47,48)(40,42,45,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,35)(34,36)(37,46)(38,43)(39,41)(40,48)(42,47)(44,45);(1,6,7,3)(2,8,4,10)(5,11,13,14)(9,16,12,15)(17,21,22,31)(18,25,23,29)(19,26,27,30)(20,28,24,32)(33,37)(34,40)(35,42)(36,44)(38,47)(39,48)(41,45)(43,46)

Polycyclic presentation:

2^9|a^2=gh,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, 10388060);