#9267027

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
prime radical	#52772	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174223);