#9240971

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	#40679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 9148167);