#9712332

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=gi,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, 9619528);