#8873605

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

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

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, 8780801);