#9210291

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

3805 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:

(1,2,7,6)(3,8,11,12)(4,13,14,9)(5,10,15,16)(17,18)(19,20)(21,22)(23,27)(24,25)(26,32)(28,29)(30,31)(35,37)(39,41)(43,45)(47,48);(1,3)(2,8)(4,5)(6,12)(7,11)(9,16)(10,13)(14,15)(17,19,23,25)(18,24,27,20)(21,30,26,28)(22,29,32,31)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(3,11)(4,14)(8,12)(9,13)(17,20,23,24)(18,25,27,19)(21,29,26,31)(22,30,32,28)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,4,7,14)(2,9,6,13)(3,5,11,15)(8,16,12,10)(17,20)(18,19)(21,31)(22,30)(23,24)(25,27)(26,29)(28,32);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,21)(18,22)(19,28)(20,29)(23,26)(24,31)(25,30)(27,32)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,6,7,2)(3,12,11,8)(4,13,14,9)(5,10,15,16)(17,22)(18,26)(19,29)(20,30)(21,27)(23,32)(24,28)(25,31)(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=h,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, 9117487);