#8618119

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	#52372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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)(3,11)(4,14)(5,12)(6,8)(7,13)(9,15)(10,16)(33,34,40,43)(35,44,46,41)(36,39,37,47)(38,42,48,45);(1,3)(2,7)(4,10)(5,14)(6,13)(8,11)(9,16)(12,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35,40,46)(34,41,43,44)(36,42,37,45)(38,47,48,39);(2,8)(5,16)(7,11)(9,14)(18,23)(20,29)(24,30)(26,32)(33,36,40,37)(34,39,43,47)(35,42,46,45)(38,44,48,41);(1,4,6,15)(2,9,8,14)(3,12,13,10)(5,11,16,7)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,37,40,36)(34,39,43,47)(35,45,46,42)(38,44,48,41);(1,5)(2,10)(3,9)(4,11)(6,16)(7,15)(8,12)(13,14)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,38)(34,42)(35,39)(36,41)(37,44)(40,48)(43,45)(46,47);(1,3)(2,11)(4,10)(5,9)(6,13)(7,8)(12,15)(14,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,39)(34,37)(35,38)(36,43)(40,47)(41,45)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=g,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, 8525315);