#8905212

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

4245 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)(17,18,22,24)(19,27,29,23)(20,25,31,30)(21,28,32,26)(34,36)(40,41)(42,44)(47,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,19,22,29)(18,23,24,27)(20,26,31,28)(21,25,32,30)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(3,11)(4,13)(7,14)(8,16)(18,24)(21,32)(23,27)(25,30)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,18)(19,23)(20,30)(21,26)(22,24)(25,31)(27,29)(28,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,20,22,31)(18,25,24,30)(19,28,29,26)(21,27,32,23)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,21)(18,26)(19,25)(20,27)(22,32)(23,31)(24,28)(29,30)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gh,e^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, 8812408);