#10202033

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,26,28,32)(34,39)(35,43)(36,44)(42,46);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,32,30,26)(33,34)(35,40)(36,41)(37,46)(38,39)(42,45)(43,47)(44,48);(2,8)(3,11)(4,13)(10,14)(17,18,23,21)(19,24,29,27)(20,25,31,30)(22,26,28,32)(33,35,38,43)(34,40,39,47)(36,45,44,37)(41,42,48,46);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,32)(26,27)(33,36,38,44)(34,41,39,48)(35,37,43,45)(40,46,47,42);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21,23,18)(19,27,29,24)(20,25,31,30)(22,26,28,32)(33,37)(34,42)(35,36)(38,45)(39,46)(40,48)(41,47)(43,44);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,25,27,30)(33,34)(35,40)(36,41)(37,46)(38,39)(42,45)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=hi,e^2=gh,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, 10109229);