#8679885

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

3929 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587081);