#10324669

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	#40963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(34,40)(35,36)(38,48)(42,45);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(19,27)(21,31)(23,32)(25,28)(33,34,39,40)(35,41,36,44)(37,38,47,48)(42,46,45,43);(5,13)(8,12)(10,16)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35,39,36)(34,41,40,44)(37,45,47,42)(38,43,48,46);(1,4,6,2)(3,7,11,9)(5,8,13,12)(10,15,16,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,39,35)(34,41,40,44)(37,45,47,42)(38,46,48,43);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37)(34,38)(35,42)(36,45)(39,47)(40,48)(41,46)(43,44);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

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

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, 10231865);