#10147567

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18)(19,25)(20,30)(21,23)(22,26)(24,27)(28,31)(29,32)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,42,44,48);(2,7)(4,8)(9,13)(12,14)(17,19,24,21)(18,25,27,23)(20,29,22,28)(26,31,30,32)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,48,46,42);(2,7)(3,10)(9,13)(11,16)(17,20,24,22)(18,26,27,30)(19,28,21,29)(23,31,25,32)(33,34,39,37)(35,40,45,43)(36,41,47,46)(38,42,44,48);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,21,24,19)(18,23,27,25)(20,28,22,29)(26,32,30,31)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,48)(42,43);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,22,24,20)(18,26,27,30)(19,29,21,28)(23,31,25,32)(33,37,39,34)(35,43,45,40)(36,41,47,46)(38,42,44,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,23)(18,19)(20,31)(21,27)(22,32)(24,25)(26,28)(29,30)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,41,43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=gi,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, 10054763);