#8726324

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

3929 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=ghi,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, 8633520);