#8375651

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	#40046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8282847);