#9210289

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

3883 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=gi,e^2=h,f^2=hi,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, 9117485);