#10171852

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,38,41)(35,40,44,39)(36,37,47,45)(42,48,46,43);(2,7)(4,15)(8,11)(10,13)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,34,38,41)(35,40,44,39)(36,45,47,37)(42,43,46,48);(2,7)(3,12)(9,14)(10,13)(18,25)(19,29)(20,31)(27,32)(33,35,38,44)(34,39,41,40)(36,42,47,46)(37,43,45,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,41)(42,47)(45,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36,38,47)(34,37,41,45)(35,42,44,46)(39,43,40,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=i,f^2=g,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, 10079048);