#8095357

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,8)(4,5)(6,7)(9,13)(10,14)(11,12)(15,16)(18,22)(20,27)(23,28)(26,31)(34,38)(35,36)(42,45)(43,44);(3,10)(8,14)(11,15)(12,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(4,9)(8,14)(11,15)(19,25)(23,28)(26,31)(30,32)(33,35,39,36)(34,40,38,41)(37,44,46,43)(42,48,45,47);(1,3,6,10)(2,8,7,14)(4,11,9,15)(5,12,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,35)(34,41,38,40)(37,44,46,43)(42,48,45,47);(1,4,6,9)(2,5,7,13)(3,11,10,15)(8,12,14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5)(2,9)(3,12)(4,7)(6,13)(8,15)(10,16)(11,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002553);