#9724827

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

4159 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18)(19,26)(20,22)(21,25)(23,27)(24,31)(28,32)(29,30)(33,34)(35,41)(36,47)(37,40)(38,42)(39,45)(43,48)(44,46);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,40,46)(34,41,37,44)(36,45,38,43)(39,42,48,47);(2,9)(4,11)(8,16)(12,14)(17,20,23,21)(18,22,27,25)(19,28,31,29)(24,30,26,32)(33,36,40,38)(34,42,37,47)(35,43,46,45)(39,44,48,41);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,37,40,34)(35,44,46,41)(36,42,38,47)(39,45,48,43);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,19)(18,26)(20,28)(21,29)(22,32)(23,31)(24,27)(25,30)(33,38,40,36)(34,42,37,47)(35,45,46,43)(39,44,48,41);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,28,26,29)(33,39,40,48)(34,43,37,45)(35,42,46,47)(36,44,38,41)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,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, 9632023);