#9717623

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

4041 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=g,f^2=gh,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, 9624819);