#10202111

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	#56313	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(18,21)(19,25)(24,28)(26,29)(33,34)(35,39)(36,40)(37,48)(38,42)(41,45)(43,44)(46,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18,22,21)(19,23,25,27)(20,28,30,24)(26,32,29,31)(33,35,38,44)(34,39,42,43)(36,45,47,37)(40,41,46,48);(17,19,22,25)(18,23,21,27)(20,26,30,29)(24,31,28,32)(33,34,38,42)(35,43,44,39)(36,40,47,46)(37,41,45,48);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,44)(45,48);(1,5,6,2)(3,9,11,7)(4,8,13,12)(10,14,16,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,36,38,47)(34,40,42,46)(35,37,44,45)(39,48,43,41);(1,3)(2,9)(4,10)(5,7)(6,11)(8,15)(12,14)(13,16)(17,21,22,18)(19,27,25,23)(20,24,30,28)(26,31,29,32)(33,37)(34,41)(35,36)(38,45)(39,46)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=ghi,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, 10109307);