#9267009

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	#2815	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=i,e^2=h,f^2=g,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, 9174205);