#8905217

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	#47002	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=g,f^2=g,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, 8812413);