#9001062

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8908258);