#10195741

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

4070 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,23,25)(19,26,22,24)(20,29,30,21)(27,32,31,28)(33,34,39,42)(35,40,38,48)(36,41,46,37)(43,44,45,47);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19,23,22)(18,24,25,26)(20,27,30,31)(21,32,29,28)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(2,8)(3,11)(4,13)(10,14)(18,25)(20,30)(24,26)(27,31)(33,36,39,46)(34,37,42,41)(35,43,38,45)(40,47,48,44);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20,23,30)(18,21,25,29)(19,27,22,31)(24,32,26,28)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21)(18,20)(19,28)(22,32)(23,29)(24,31)(25,30)(26,27)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,22,23,19)(18,26,25,24)(20,31,30,27)(21,28,29,32)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,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, 10102937);