#8978372

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=g,f^2=gh,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, 8885568);