#9717594

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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)(3,8)(4,7)(5,9)(6,14)(10,16)(11,12)(13,15)(17,18)(19,29)(20,21)(22,26)(23,31)(24,28)(25,32)(27,30)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,47)(45,48);(1,3)(2,8)(4,12)(5,14)(6,9)(7,11)(10,15)(13,16)(17,19)(18,25)(20,30)(21,31)(22,23)(24,32)(26,27)(28,29)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(2,4)(3,11)(9,13)(10,14)(17,20,24,22)(18,26,28,21)(19,30,32,23)(25,27,29,31)(33,36,38,45)(34,40,42,48)(35,44,46,37)(39,41,43,47);(1,4,7,2)(3,12,11,8)(5,13,15,9)(6,14,16,10)(17,21,24,26)(18,20,28,22)(19,31,32,27)(23,25,30,29);(1,5,7,15)(2,9,4,13)(3,10,11,14)(6,8,16,12)(17,22,24,20)(18,26,28,21)(19,30,32,23)(25,31,29,27)(33,37,38,44)(34,41,42,47)(35,45,46,36)(39,40,43,48);(1,6)(2,10)(3,9)(4,14)(5,12)(7,16)(8,15)(11,13)(17,23,24,30)(18,27,28,31)(19,22,32,20)(21,25,26,29)(33,34)(35,39)(36,40)(37,47)(38,42)(41,44)(43,46)(45,48)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,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, 9624790);