#8137612

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044808);