#9716824

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3815 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)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34)(35,43)(36,37)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3,7,12)(2,8,9,16)(4,13,15,6)(5,11,10,14)(17,19,23,31)(18,24,28,29)(20,30,21,22)(25,27,26,32)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43);(2,9)(3,12)(4,15)(11,14)(17,20,23,21)(18,25,28,26)(19,30,31,22)(24,27,29,32)(33,36,39,38)(34,41,42,37)(35,44,46,47)(40,48,43,45);(1,4)(2,10)(3,13)(5,9)(6,12)(7,15)(8,14)(11,16)(17,21,23,20)(18,26,28,25)(19,22,31,30)(24,32,29,27)(33,37,39,41)(34,36,42,38)(35,45,46,48)(40,44,43,47);(1,5,7,10)(2,4,9,15)(3,14,12,11)(6,16,13,8)(17,22)(18,27)(19,20)(21,31)(23,30)(24,26)(25,29)(28,32)(33,38,39,36)(34,41,42,37)(35,44,46,47)(40,45,43,48);(1,6,7,13)(2,11,9,14)(3,4,12,15)(5,8,10,16)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=h,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, 9624020);