#9015649

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	#55161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,23,28)(19,26,21,24)(20,30,31,25)(22,27,29,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(17,19,23,21)(18,24,28,26)(20,22,31,29)(25,27,30,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(2,9)(3,6)(4,14)(5,15)(17,20,23,31)(18,25,28,30)(19,29,21,22)(24,32,26,27)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,21,23,19)(18,26,28,24)(20,22,31,29)(25,27,30,32)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,22)(18,27)(19,20)(21,31)(23,29)(24,25)(26,30)(28,32)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922845);