#8375648

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	#42541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,23,27)(19,24,28,31)(20,21,25,32)(22,26,29,30)(34,39)(40,43)(41,46)(47,48);(2,7)(3,10)(4,13)(15,16)(17,19,23,28)(18,24,27,31)(20,29,25,22)(21,30,32,26)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(2,7)(5,8)(9,12)(15,16)(19,28)(22,29)(24,31)(26,30)(33,35,38,36)(34,40,39,43)(37,44,45,42)(41,48,46,47);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,20)(18,21)(19,29)(22,28)(23,25)(24,30)(26,31)(27,32)(33,34)(35,40)(36,43)(37,41)(38,39)(42,47)(44,48)(45,46);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,21)(18,25)(19,26)(20,27)(22,31)(23,32)(24,29)(28,30)(33,36,38,35)(34,40,39,43)(37,44,45,42)(41,47,46,48);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,22,23,29)(18,26,27,30)(19,20,28,25)(21,31,32,24)(33,37,38,45)(34,41,39,46)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 8282844);