#10143640

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	#58087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(3,10)(4,13)(5,12)(6,8)(7,14)(9,15)(11,16)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,47)(46,48);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27)(33,35,40,39)(34,41,43,44)(36,45,37,46)(38,47,42,48);(2,8)(4,15)(7,10)(11,12)(18,25)(19,28)(20,30)(27,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,48,44,47);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,37,40,36)(34,42,43,38)(35,46,39,45)(41,48,44,47);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31)(33,38,40,42)(34,37,43,36)(35,47,39,48)(41,46,44,45);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,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, 10050836);