#10201910

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	#55169	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=gi,e^2=gh,f^2=hi,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, 10109106);