#9240933

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=h,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, 9148129);