#9851259

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3750 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758455);