#10196170

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	#57375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=gi,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, 10103366);