#9712446

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	#53876	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(3,6)(4,5)(7,8)(9,13)(10,16)(11,14)(12,15)(17,18,22,26)(19,25,29,23)(20,21,30,24)(27,31,32,28)(36,44)(39,46)(43,48)(45,47);(1,3,7,10)(2,6,8,16)(4,11,13,15)(5,14,9,12)(17,19,22,29)(18,23,26,25)(20,27,30,32)(21,28,24,31)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(2,8)(3,10)(4,13)(12,14)(17,20,22,30)(18,24,26,21)(19,27,29,32)(23,31,25,28)(33,35)(34,37)(36,43)(38,41)(39,45)(40,42)(44,48)(46,47);(1,4)(2,9)(3,11)(5,8)(6,12)(7,13)(10,15)(14,16)(17,20)(18,21)(19,27)(22,30)(23,28)(24,26)(25,31)(29,32)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,48)(42,46);(1,5,7,9)(2,4,8,13)(3,12,10,14)(6,15,16,11)(17,19)(18,25)(20,27)(21,31)(22,29)(23,26)(24,28)(30,32)(33,35)(34,40)(36,43)(37,42)(38,41)(39,47)(44,48)(45,46);(1,6)(2,10)(3,8)(4,12)(5,11)(7,16)(9,15)(13,14)(17,21)(18,20)(19,28)(22,24)(23,27)(25,32)(26,30)(29,31)(33,37,38,40)(34,41,42,35)(36,45,44,47)(39,48,46,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=gh,f^2=ghi,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, 9619642);