#9267004

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	#54751	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=gi,e^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, 9174200);