#9719932

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	#53444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,31,29)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,30)(18,24,27,28)(20,21,26,31)(22,29,32,25)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,6)(7,13)(8,11)(14,16)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,26)(18,22,27,32)(19,21,30,31)(24,29,28,25)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21)(18,25)(19,26)(20,30)(22,24)(23,31)(27,29)(28,32)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,26)(19,29)(20,27)(21,28)(23,32)(24,31)(25,30)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=i,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, 9627128);