#8734425

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,22,25)(19,24,28,23)(20,29,30,21)(26,27,32,31)(34,38)(36,43)(39,44)(42,47);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(35,41)(39,44)(42,47)(46,48);(3,11)(4,12)(7,16)(8,13)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,19)(18,24)(20,26)(21,31)(22,28)(23,25)(27,29)(30,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,20,22,30)(18,21,25,29)(19,26,28,32)(23,27,24,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,21)(18,20)(19,27)(22,29)(23,26)(24,32)(25,30)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8641621);