#10183149

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	#53395	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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)(17,18,22,24)(19,26,21,23)(20,25,30,29)(27,32,28,31)(33,34,40,44)(35,41,38,43)(36,42,37,39)(45,46,47,48);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,22,21)(18,23,24,26)(20,28,30,27)(25,32,29,31)(33,35,40,38)(34,41,44,43)(36,47,37,45)(39,46,42,48);(2,9)(3,11)(4,13)(10,14)(18,24)(23,26)(25,29)(31,32)(33,36,40,37)(34,42,44,39)(35,45,38,47)(41,46,43,48);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,32)(28,31)(33,37,40,36)(34,39,44,42)(35,45,38,47)(41,46,43,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,20,22,30)(18,25,24,29)(19,27,21,28)(23,31,26,32)(33,38,40,35)(34,43,44,41)(36,47,37,45)(39,46,42,48);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,21,22,19)(18,26,24,23)(20,27,30,28)(25,31,29,32)(33,39)(34,36)(35,46)(37,44)(38,48)(40,42)(41,47)(43,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,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, 10090345);