#8479552

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

3930 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8386748);