#10038420

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,8,11,14)(4,9,10,5)(12,15,16,13)(19,25)(23,26)(27,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(2,7)(3,11)(4,10)(13,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(34,38)(39,45)(40,46)(47,48);(2,7)(3,11)(4,10)(13,15)(18,22)(23,26)(24,28)(30,32)(33,35,37,42)(34,39,38,45)(36,41,44,43)(40,47,46,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,15)(10,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4)(2,9)(3,12)(5,7)(6,10)(8,15)(11,16)(13,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,35)(34,39)(36,43)(37,42)(38,45)(40,48)(41,44)(46,47);(1,5,6,9)(2,10,7,4)(3,13,11,15)(8,16,14,12)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36)(34,40)(35,41)(37,44)(38,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9945616);