#8902463

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8809659);