#10368642

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275838);