#9712065

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

3797 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619261);