#9945577

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

4250 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)(3,4)(5,8)(6,14)(7,9)(10,15)(11,16)(12,13)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,9)(4,8)(10,14)(12,16)(18,25)(19,29)(20,31)(27,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,7,3)(2,8,9,4)(6,11,15,13)(10,12,14,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,6,7,15)(2,10,9,14)(3,11,5,13)(4,12,8,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9852773);