#9717669

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

4187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=ghi,f^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, 9624865);