#8772663

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679859);