#9717511

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

4027 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,20)(21,25)(22,24)(23,27)(26,28)(29,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,48)(44,47)(45,46);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,35)(34,38)(36,42)(37,45)(39,41)(40,46)(43,47)(44,48);(2,4)(3,10)(9,12)(11,16)(18,24)(20,23)(25,32)(28,31)(34,36)(35,41)(40,44)(43,45);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,36,39,34)(35,42,41,38)(37,44,47,40)(43,46,45,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37,39,47)(34,40,36,44)(35,43,41,45)(38,48,42,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,38,39,42)(34,41,36,35)(37,46,47,48)(40,43,44,45)

Polycyclic presentation:

2^9|d^2=h,e^2=gh,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, 9624707);