#8977348

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	#53681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(33,34,38,43)(35,39,46,48)(36,41,45,40)(37,42,44,47);(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)(33,35,38,46)(34,39,43,48)(36,44,45,37)(40,42,41,47);(3,9)(4,8)(5,13)(10,14)(18,23)(19,27)(20,30)(21,31)(33,36,38,45)(34,40,43,41)(35,37,46,44)(39,47,48,42);(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)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,43)(42,48);(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)(33,37,38,44)(34,42,43,47)(35,45,46,36)(39,40,48,41);(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)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gh,f^2=g,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, 8884544);