#9053976

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,39)(36,42)(37,41)(38,40)(43,45)(44,48)(46,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(20,28)(23,30)(26,32)(29,31)(34,35)(37,45)(40,42)(44,46);(1,2,6,9)(3,7,4,10)(5,13,14,8)(11,16,12,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35,39,34)(36,42,38,40)(37,43,45,41)(44,48,46,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(33,36,39,38)(34,40,35,42)(37,44,45,46)(41,47,43,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37,39,45)(34,41,35,43)(36,44,38,46)(40,47,42,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,38,39,36)(34,42,35,40)(37,46,45,44)(41,48,43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,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, 8961172);