#9715453

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	#49751	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=g,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, 9622649);