#8734920

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	#53066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8642116);