#10195769

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	#48063	C₂
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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=gh,e^2=gh,f^2=hi,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, 10102965);