#10506282

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

4112 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 10413478);