#8905222

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(17,18,22,26)(19,21,25,23)(20,24,30,32)(27,29,28,31)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(34,37)(35,42)(41,45)(43,48);(3,11)(4,13)(7,14)(8,16)(19,25)(21,23)(27,28)(29,31)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,29,23,31)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=gh,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, 8812418);