#8425666

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

3978 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18)(19,22)(20,30)(21,29)(23,28)(24,27)(25,32)(26,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(3,4)(7,10)(11,12)(15,16)(17,19,23,27)(18,24,28,22)(20,29,32,26)(21,30,31,25)(35,42)(39,46)(43,47)(44,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,20)(18,25)(19,29)(21,24)(22,31)(23,32)(26,27)(28,30)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,31)(18,26,28,29)(19,25,27,30)(20,24,32,22)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22,23,24)(18,27,28,19)(20,31,32,21)(25,26,30,29);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,29)(22,27)(23,28)(26,31)(30,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=h,e^2=h,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, 8332862);