#10202104

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	#57540	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,10)(4,13)(5,12)(6,8)(7,14)(9,15)(11,16)(17,18)(19,29)(20,26)(21,22)(23,27)(24,28)(25,31)(30,32)(33,34,38,41)(35,40,44,39)(36,37,47,45)(42,48,46,43);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19)(18,25)(20,23)(21,30)(22,27)(24,31)(26,32)(28,29)(33,34,38,41)(35,40,44,39)(36,45,47,37)(42,43,46,48);(2,8)(4,15)(7,10)(11,12)(17,20,24,21)(18,26,28,22)(19,23,31,30)(25,32,29,27)(33,35,38,44)(34,39,41,40)(36,42,47,46)(37,43,45,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,21,24,20)(18,26,28,22)(19,30,31,23)(25,32,29,27)(33,35)(34,40)(36,46)(37,43)(38,44)(39,41)(42,47)(45,48);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,22,24,26)(18,21,28,20)(19,27,31,32)(23,25,30,29)(33,36,38,47)(34,37,41,45)(35,42,44,46)(39,43,40,48);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,23,24,30)(18,27,28,32)(19,20,31,21)(22,29,26,25)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=ghi,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, 10109300);