#8919296

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	#40426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=hi,f^2=h,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, 8826492);