#8157361

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	#53408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8064557);