#8745836

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3822 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=ghi,f^2=gh,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, 8653032);