#9716235

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

3867 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9623431);