#9713322

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=g,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, 9620518);