#9481428

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

4047 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)(6,9)(7,11)(10,12)(13,15)(14,16)(17,18)(19,24)(20,30)(21,27)(22,31)(23,25)(26,29)(28,32)(35,41)(39,46)(42,44)(47,48);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,19,23,28)(18,24,25,32)(20,21,29,31)(22,30,27,26)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(5,13)(8,15)(10,16)(12,14)(18,25)(19,28)(20,29)(22,27)(34,38)(39,46)(40,43)(47,48);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,20)(18,26)(19,21)(22,24)(23,29)(25,30)(27,32)(28,31)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,21)(18,27)(19,29)(20,28)(22,25)(23,31)(24,26)(30,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,27)(18,21,25,31)(19,30,28,26)(20,24,29,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,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, 9388624);