#8534891

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	#36987	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442087);