#8734909

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

3992 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)(3,5)(4,8)(6,9)(7,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,42)(35,43,44,39)(36,40,47,45)(37,41,46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,34)(35,39)(36,45)(37,41)(38,42)(40,47)(43,44)(46,48);(4,12)(8,16)(10,14)(13,15)(19,27)(20,28)(23,32)(24,29)(33,35,38,44)(34,39,42,43)(36,46,47,37)(40,41,45,48);(1,4,6,12)(2,8,9,16)(3,10,11,14)(5,13,7,15)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36,38,47)(34,40,42,45)(35,37,44,46)(39,48,43,41);(1,5,6,7)(2,3,9,11)(4,13,12,15)(8,10,16,14)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,34)(35,39)(36,40)(37,48)(38,42)(41,46)(43,44)(45,47);(1,3)(2,5)(4,14)(6,11)(7,9)(8,15)(10,12)(13,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37)(34,41)(35,36)(38,46)(39,45)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8642105);