#8718974

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	#43362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4179 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,31)(29,30)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,7)(3,12)(9,14)(10,13)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,32,31)(34,35)(39,41)(40,42)(46,47);(2,7)(3,12)(9,14)(10,13)(18,25)(19,27)(20,26)(29,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,29,24,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,26)(19,29)(20,25)(22,30)(23,32)(24,28)(27,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22,23,24)(18,27,25,19)(20,31,26,29)(21,28,32,30)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=i,e^2=g,f^2=h,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, 8626170);