#10171855

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

4054 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,40,43)(35,44,47,41)(36,38,37,48)(39,42,45,46);(2,7)(4,10)(9,13)(12,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35)(34,41)(36,39)(37,45)(38,46)(40,47)(42,48)(43,44);(2,7)(3,8)(9,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,36,40,37)(34,38,43,48)(35,39,47,45)(41,46,44,42);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37,40,36)(34,38,43,48)(35,45,47,39)(41,46,44,42);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,38)(34,37)(35,46)(36,43)(39,44)(40,48)(41,45)(42,47);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,39,40,45)(34,42,43,46)(35,36,47,37)(38,41,48,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=ghi,e^2=i,f^2=g,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, 10079051);