#9245123

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	#58177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,4,5,9)(6,14,15,10)(11,16,13,12)(17,18,22,21)(19,23,27,26)(20,28,29,24)(25,31,32,30)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,8)(3,5)(10,14)(11,13)(18,21)(23,26)(24,28)(30,31)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,3,7,5)(2,9,8,4)(6,11,15,13)(10,16,14,12)(17,19,22,27)(18,23,21,26)(20,25,29,32)(24,30,28,31)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4,7,9)(2,3,8,5)(6,12,15,16)(10,11,14,13)(17,20)(18,24)(19,25)(21,28)(22,29)(23,30)(26,31)(27,32);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,18)(19,26,27,23)(20,28,29,24)(25,31,32,30)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,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, 9152319);