#8720441

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627637);