#8425654

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332850);