#10183123

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=hi,e^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, 10090319);