#9058522

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	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965718);