#8618294

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525490);