#10423073

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	#50740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34,40,44)(35,43,39,36)(37,46,48,41)(38,45,47,42);(2,4)(7,9)(8,12)(15,16)(19,23)(20,27)(26,31)(28,32)(33,35,40,39)(34,36,44,43)(37,45,48,42)(38,41,47,46);(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,36)(34,39)(35,44)(37,47)(38,48)(40,43)(41,42)(45,46);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,37,40,48)(34,41,44,46)(35,45,39,42)(36,47,43,38);(1,4,6,2)(3,7,11,9)(5,12,14,8)(10,15,13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,38,40,47)(34,42,44,45)(35,46,39,41)(36,48,43,37);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,39,40,35)(34,43,44,36)(37,45,48,42)(38,41,47,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10330269);