#9545666

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	#55605	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9452862);