#10201930

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,22,23)(19,27,28,24)(20,25,29,32)(21,30,31,26)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,47)(45,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(18,23)(20,29)(24,27)(26,30)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,47,46,48);(17,19,22,28)(18,24,23,27)(20,26,29,30)(21,25,31,32)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,47,41,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19)(18,24)(20,30)(21,25)(22,28)(23,27)(26,29)(31,32)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,48,46,47);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,20)(18,25)(19,26)(21,27)(22,29)(23,32)(24,31)(28,30)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,47,41,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,21,22,31)(18,26,23,30)(19,25,28,32)(20,24,29,27)(33,39)(34,35)(36,47)(37,43)(38,48)(40,41)(42,44)(45,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,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, 10109126);