#9210377

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=i,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=dgi,e^d=egi,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, 9117573);