#8873579

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(34,36)(37,43)(41,44)(45,48);(2,7)(4,8)(5,14)(11,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(33,34,39,36)(35,37,40,43)(38,44,46,41)(42,48,47,45);(2,7)(3,9)(10,13)(11,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35,39,40)(34,37,36,43)(38,42,46,47)(41,45,44,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36,39,34)(35,37,40,43)(38,44,46,41)(42,45,47,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,37)(34,40)(35,36)(38,45)(39,43)(41,47)(42,44)(46,48);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=ghi,e^2=hi,f^2=h,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, 8780775);