#9910957

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=h,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, 9818153);