#8461609

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8368805);