#8726211

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(3,11,12,8)(4,13,14,9)(5,10,15,16)(33,34,39,43)(35,38,42,40)(36,41,46,37)(44,45,48,47);(2,7)(3,12)(9,13)(10,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,35,39,42)(34,40,43,38)(36,44,46,48)(37,45,41,47);(2,7)(3,12)(9,13)(10,16)(18,22)(23,30)(24,27)(31,32)(33,36,39,46)(34,41,43,37)(35,44,42,48)(38,45,40,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31)(33,36)(34,37)(35,44)(38,47)(39,46)(40,45)(41,43)(42,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,37)(34,36)(35,45)(38,48)(39,41)(40,44)(42,47)(43,46);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32)(33,38,39,40)(34,42,43,35)(36,45,46,47)(37,44,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633407);