#8750759

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

3743 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=h,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, 8657955);