#10202005

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

4072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=hi,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, 10109201);