#8534897

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

3999 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)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(17,18,22,24)(19,27,28,23)(20,29,31,25)(21,26,30,32);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,22,28)(18,23,24,27)(20,30,31,21)(25,32,29,26)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(5,13)(8,12)(10,16)(14,15)(18,24)(23,27)(25,29)(26,32)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,31)(18,25,24,29)(19,21,28,30)(23,26,27,32)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,21)(18,26)(19,20)(22,30)(23,25)(24,32)(27,29)(28,31)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8442093);