#10202019

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

4212 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)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,9)(3,11)(4,13)(10,14)(18,25)(20,27)(24,32)(28,30)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,21,23,26)(18,27,25,20)(19,29,22,31)(24,28,32,30)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,22,23,19)(18,24,25,32)(20,30,27,28)(21,31,26,29)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=ghi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10109215);