#10521309

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

4165 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=gi,f^2=g,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, 10428505);