#10171378

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	#47889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(18,19)(23,25)(24,26)(31,32)(33,34)(35,40)(36,38)(37,42)(39,41)(43,45)(44,47)(46,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,35,39,36)(34,40,41,38)(37,44,46,43)(42,47,48,45);(2,6)(7,13)(8,11)(14,16)(34,41)(35,36)(42,48)(43,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,36,39,35)(34,38,41,40)(37,43,46,44)(42,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,46)(34,42,41,48)(35,43,36,44)(38,47,40,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,38)(34,35)(36,41)(37,45)(39,40)(42,44)(43,48)(46,47)

Polycyclic presentation:

2^9|b^2=gi,d^2=gi,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, 10078574);