#9862990

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770186);