#9923900

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

4013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=h,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, 9831096);