#8876047

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8783243);