#8919409

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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)(3,11)(4,6)(5,8)(7,10)(9,14)(12,16)(13,15)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,3,7,5)(2,8,10,11)(4,12,14,15)(6,13,9,16)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(34,39)(36,42)(41,45)(44,48);(4,14)(6,9)(12,15)(13,16)(18,23)(19,26)(21,30)(27,29)(33,35,38,40)(34,36,39,42)(37,46,47,43)(41,48,45,44);(1,4)(2,9)(3,12)(5,15)(6,10)(7,14)(8,16)(11,13)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,5,7,3)(2,8,10,11)(4,12,14,15)(6,16,9,13)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36)(34,40)(35,39)(37,44)(38,42)(41,46)(43,45)(47,48);(1,6,7,9)(2,4,10,14)(3,13,5,16)(8,12,11,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37)(34,41)(35,43)(36,44)(38,47)(39,45)(40,46)(42,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8826605);