#10171910

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	#54037	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=ghi,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, 10079106);