#9712330

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3836 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:

(1,2,6,8)(3,9,13,7)(4,14,10,5)(11,12,16,15)(18,24)(20,25)(26,31)(28,32);(1,3,6,13)(2,7,8,9)(4,11,10,16)(5,12,14,15)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34,39,38)(35,43,45,40)(36,41,44,46)(37,42,47,48);(2,8)(4,10)(7,9)(11,16)(17,19,23,21)(18,20,24,25)(22,27,30,29)(26,28,31,32)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(1,4,6,10)(2,5,8,14)(3,11,13,16)(7,12,9,15)(17,20,23,25)(18,19,24,21)(22,28,30,32)(26,27,31,29)(33,36,39,44)(34,41,38,46)(35,37,45,47)(40,48,43,42);(1,3)(2,9)(4,11)(5,15)(6,13)(7,8)(10,16)(12,14)(17,21,23,19)(18,25,24,20)(22,27,30,29)(26,28,31,32)(33,37)(34,42)(35,44)(36,45)(38,48)(39,47)(40,41)(43,46);(1,5)(2,10)(3,12)(4,8)(6,14)(7,16)(9,11)(13,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(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=hi,c^2=ghi,d^2=gi,e^2=gh,f^2=i,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, 9619526);