#8687327

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

3749 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594523);