#9712435

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,23,27)(19,28,30,24)(20,25,31,21)(22,29,32,26)(36,44)(39,46)(42,48)(45,47);(1,3,7,12)(2,8,9,16)(4,13,15,6)(5,11,10,14)(17,19,23,30)(18,24,27,28)(20,29,31,26)(21,32,25,22)(33,34)(35,40)(36,39)(37,41)(38,43)(42,47)(44,46)(45,48);(2,9)(3,12)(4,15)(11,14)(17,20,23,31)(18,21,27,25)(19,29,30,26)(22,24,32,28)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4)(2,10)(3,13)(5,9)(6,12)(7,15)(8,14)(11,16)(17,20)(18,25)(19,29)(21,27)(22,24)(23,31)(26,30)(28,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,47)(41,46)(43,48);(1,5,7,10)(2,4,9,15)(3,14,12,11)(6,16,13,8)(17,21,23,25)(18,20,27,31)(19,22,30,32)(24,26,28,29)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6,7,13)(2,11,9,14)(3,4,12,15)(5,8,10,16)(17,22)(18,26)(19,21)(20,24)(23,32)(25,30)(27,29)(28,31)(33,34)(35,38)(36,39)(37,41)(40,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,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, 9619631);