#10168362

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	#55251	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(18,22)(20,28)(23,25)(26,32)(34,39)(35,43)(36,44)(42,46);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,40)(36,41)(37,46)(38,39)(42,45)(43,47)(44,48);(2,9)(3,4)(10,14)(11,12)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,35,38,43)(34,40,39,47)(36,45,44,37)(41,42,48,46);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,44)(34,41,39,48)(35,37,43,45)(40,46,47,42);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,42)(35,36)(38,45)(39,46)(40,48)(41,47)(43,44);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,40)(36,41)(37,46)(38,39)(42,45)(43,47)(44,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,e^2=ghi,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, 10075558);