#10195405

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

3839 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=g,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, 10102601);