#8749621

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

3869 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)(4,8,14,12)(10,16,13,15)(17,18,23,24)(19,28,22,25)(20,26,30,32)(21,29,31,27)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,2,5,6)(3,7,11,9)(4,8,14,12)(10,15,13,16)(18,24)(20,30)(25,28)(27,29)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,6)(7,9)(8,12)(15,16)(17,19,23,22)(18,25,24,28)(20,27,30,29)(21,26,31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20,23,30)(18,26,24,32)(19,29,22,27)(21,28,31,25)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,27)(19,26)(20,28)(22,32)(23,31)(24,29)(25,30)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22,23,19)(18,25,24,28)(20,29,30,27)(21,26,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8656817);