#10173519

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

4170 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=ghi,e^2=ghi,f^2=gh,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, 10080715);