#8649852

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

3888 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)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(33,34,40,42)(35,36,38,39)(37,41,46,48)(43,44,45,47);(2,7)(3,11)(4,14)(13,15)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30)(33,35)(34,36)(37,45)(38,40)(39,42)(41,47)(43,46)(44,48);(2,7)(4,14)(8,10)(12,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,32,31)(33,36,40,39)(34,35,42,38)(37,44,46,47)(41,43,48,45);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,20,22,19)(18,23,25,26)(21,27,29,28)(24,31,32,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,47)(40,46)(42,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32)(33,38,40,35)(34,36,42,39)(37,45,46,43)(41,44,48,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21,22,29)(18,24,25,32)(19,27,20,28)(23,30,26,31)(33,39,40,36)(34,35,42,38)(37,44,46,47)(41,45,48,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8557048);