#9923674

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

4392 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830870);