#9923999

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
prime radical	#55246	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=i,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, 9831195);