#8550913

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

3809 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458109);