#9713679

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,24)(20,23)(21,25)(22,30)(26,32)(27,28)(29,31)(33,34)(35,38)(36,37)(39,47)(40,42)(41,48)(43,44)(45,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,24)(20,28)(21,30)(22,25)(23,27)(26,31)(29,32)(33,35,40,37)(34,36,42,38)(39,43,48,45)(41,44,47,46);(2,3)(7,9)(8,10)(14,15)(18,20)(19,27)(25,29)(26,30)(33,36,40,38)(34,35,42,37)(39,44,48,46)(41,43,47,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,18)(19,28,27,24)(21,29,31,25)(22,30,32,26)(33,37,40,35)(34,36,42,38)(39,45,48,43)(41,44,47,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,31)(18,25,20,29)(19,26,27,30)(22,24,32,28)(33,38,40,36)(34,35,42,37)(39,44,48,46)(41,45,47,43);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,26)(19,25)(20,30)(21,28)(23,32)(24,31)(27,29)(33,39)(34,41)(35,43)(36,44)(37,45)(38,46)(40,48)(42,47)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=h,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, 9620875);