#8734927

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

3832 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=h,f^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, 8642123);