#9245129

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3710 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)(3,9)(4,8)(5,10)(6,7)(11,15)(12,14)(13,16)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,7)(4,12)(5,9)(11,16)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(34,38)(39,41)(40,45)(46,47);(2,7)(3,10)(4,12)(13,15)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,16)(14,15)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,6,12)(2,8,7,14)(3,11,10,16)(5,13,9,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=gh,e^2=h,f^2=gh,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, 9152325);