#9718416

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	#55041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=h,f^2=i,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, 9625612);