#8445861

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

3902 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(34,40)(35,43)(36,45)(42,47);(1,3)(2,8)(4,6)(5,13)(7,14)(9,10)(11,12)(15,16)(19,29)(20,30)(24,28)(25,32)(33,34,39,40)(35,38,43,48)(36,46,45,41)(37,42,44,47);(1,4,7,5)(2,9,11,15)(3,6,14,13)(8,10,12,16)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35,39,43)(34,38,40,48)(36,44,45,37)(41,42,46,47);(1,5,7,4)(2,9,11,15)(3,13,14,6)(8,10,12,16)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36)(34,41)(35,44)(37,43)(38,42)(39,45)(40,46)(47,48);(1,6,7,13)(2,10,11,16)(3,4,14,5)(8,9,12,15)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32)(33,37,39,44)(34,42,40,47)(35,36,43,45)(38,46,48,41);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,38)(34,43)(35,40)(36,47)(37,46)(39,48)(41,44)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353057);