#9240973

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	#37473	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(34,36)(39,41)(40,44)(46,47);(2,7)(4,8)(5,14)(10,16)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3,6,11)(2,8,7,4)(5,10,14,16)(9,15,13,12)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(1,5)(2,9)(3,10)(4,12)(6,14)(7,13)(8,15)(11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,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, 9148169);