#8678988

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586184);