#8618056

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

3884 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(33,34,40,44)(35,39,43,48)(36,38,47,41)(37,42,46,45);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35,40,43)(34,39,44,48)(36,42,47,45)(37,38,46,41);(2,8)(5,10)(9,12)(14,16)(18,23)(20,29)(24,30)(26,32)(33,36,40,47)(34,41,44,38)(35,45,43,42)(37,39,46,48);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,37)(34,42)(35,41)(36,39)(38,43)(40,46)(44,45)(47,48);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,38,40,41)(34,36,44,47)(35,37,43,46)(39,45,48,42);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,39)(34,43)(35,44)(36,46)(37,47)(38,45)(40,48)(41,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8525252);