#10181216

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	#56383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,47)(46,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31)(33,35,40,39)(34,41,43,44)(36,45,37,46)(38,47,42,48);(2,9)(3,13)(4,15)(11,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,48,44,47);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,37,40,36)(34,42,43,38)(35,46,39,45)(41,48,44,47);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,21,23,18)(19,26,20,24)(22,25,30,29)(27,31,28,32)(33,38,40,42)(34,37,43,36)(35,47,39,48)(41,46,44,45);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,47,42,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=gi,e^2=ghi,f^2=g,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, 10088412);