#10216583

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	#49985	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=hi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10123779);