#8137582

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	#49801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044778);