#8751487

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

3805 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)(3,4,9,12)(5,8,15,16)(10,11,13,14)(17,18,21,22)(19,25,27,23)(20,24,30,28)(26,32,29,31)(33,34,39,42)(35,37,40,36)(38,46,47,41)(43,44,48,45);(2,7)(3,9)(8,16)(10,13)(17,18,21,22)(19,23,27,25)(20,24,30,28)(26,31,29,32)(33,35,39,40)(34,36,42,37)(38,43,47,48)(41,44,46,45);(2,7)(3,9)(8,16)(10,13)(18,22)(23,25)(24,28)(31,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,3)(2,4)(5,13)(6,9)(7,12)(8,14)(10,15)(11,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,37,39,36)(34,35,42,40)(38,45,47,44)(41,43,46,48);(1,4)(2,3)(5,14)(6,12)(7,9)(8,13)(10,16)(11,15)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46);(1,5,6,15)(2,8,7,16)(3,10,9,13)(4,11,12,14)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658683);