#8733737

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=gh,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, 8640933);