#9481402

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

4067 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388598);