#8726293

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

3960 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633489);