#8188213

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	#45024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(17,18,22,23)(19,21,25,30)(20,24,29,27)(26,28,31,32)(34,36)(35,41)(40,44)(43,47);(2,9)(6,10)(8,11)(13,16)(18,23)(21,30)(24,27)(28,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095409);