#9245121

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(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);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(34,38)(39,41)(40,45)(46,47);(2,9)(3,4)(10,16)(11,12)(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);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(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=h,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, 9152317);