#8733839

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	#54811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(36,44)(41,46)(43,45)(47,48);(3,11)(7,8)(12,16)(13,14)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,36,39,44)(34,41,38,46)(35,43,37,45)(40,47,42,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 8641035);