#9712146

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

3801 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)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,18,22,24)(19,26,21,23)(20,25,29,31)(27,32,28,30)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,22,21)(18,23,24,26)(20,28,29,27)(25,32,31,30)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(18,24)(23,26)(25,31)(30,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,5,7,2)(3,9,4,8)(6,10,14,13)(11,15,12,16)(17,20,22,29)(18,25,24,31)(19,27,21,28)(23,30,26,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21,22,19)(18,26,24,23)(20,27,29,28)(25,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619342);