#10506281

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,e^2=ghi,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, 10413477);