#9053986

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	#55797	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961182);