#9685582

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9592778);