#9717513

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

4175 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,30)(28,32)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(4,11)(8,14)(10,15)(12,16)(18,23)(20,28)(24,30)(26,29)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(2,7)(3,9)(8,14)(10,15)(18,23)(19,27)(25,32)(26,29)(34,36)(35,42)(41,44)(43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,27)(18,24,23,30)(20,26,28,29)(21,25,31,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,25)(19,26)(21,30)(22,28)(23,32)(24,31)(27,29)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,31)(18,26,23,29)(19,25,27,32)(20,24,28,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|d^2=h,e^2=gh,f^2=gi,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, 9624709);