#8436927

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

(1,2,6,10)(3,5,9,7)(4,8,13,16)(11,12,15,14)(17,18)(19,29)(20,26)(21,23)(22,31)(24,28)(25,32)(27,30)(33,34,38,39)(35,36,43,40)(37,41,46,47)(42,44,48,45);(1,3,6,9)(2,5,10,7)(4,11,13,15)(8,12,16,14)(17,19)(18,25)(20,22)(21,30)(23,31)(24,32)(26,27)(28,29)(34,39)(36,40)(41,47)(44,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,24,21)(18,26,28,23)(19,22,32,30)(25,27,29,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,45)(44,47)(46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,21,24,20)(18,26,28,23)(19,30,32,22)(25,27,29,31)(33,36,38,40)(34,35,39,43)(37,45,46,44)(41,48,47,42);(1,4,6,13)(2,8,10,16)(3,11,9,15)(5,12,7,14)(17,22,24,30)(18,27,28,31)(19,20,32,21)(23,25,26,29);(1,5)(2,9)(3,10)(4,12)(6,7)(8,15)(11,16)(13,14)(17,23,24,26)(18,21,28,20)(19,31,32,27)(22,25,30,29)(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=ghi,d^2=g,e^2=gh,f^2=g,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, 8344123);