#9910219

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

4251 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817415);