#9717727

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

4128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624923);