#10482509

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

4265 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389705);