#8125483

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8032679);