#9910202

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

4413 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)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,27)(20,22)(21,24)(23,25)(26,32)(28,31)(29,30)(33,34)(35,37)(36,41)(38,42)(39,40)(43,45)(44,46)(47,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,21)(18,24,25,27)(20,28,26,30)(22,29,32,31)(34,40)(37,41)(42,48)(45,46);(2,3)(7,9)(8,10)(14,15)(18,25)(20,26)(24,27)(28,30)(33,35,39,36)(34,37,40,41)(38,43,47,44)(42,45,48,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,26)(18,22,25,32)(19,28,21,30)(24,29,27,31)(33,36,39,35)(34,41,40,37)(38,44,47,43)(42,46,48,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,24,25,27)(20,30,26,28)(22,29,32,31)(33,37,39,41)(34,35,40,36)(38,46,47,45)(42,44,48,43);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,26)(19,29)(20,25)(21,31)(23,32)(24,30)(27,28)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^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, 9817398);