#8455739

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

3896 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)(3,9)(8,12)(11,13)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(34,39)(35,40)(41,46)(42,47);(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8362935);