#8119619

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026815);