#8747906

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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)(4,5)(6,8)(7,9)(10,15)(12,13)(14,16)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,6)(2,8,9,11)(4,14,10,12)(5,13,15,16)(18,22)(19,26)(24,28)(27,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,9)(4,10)(8,11)(12,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,10)(2,5,9,15)(3,12,6,14)(8,16,11,13)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5)(2,10)(3,13)(4,9)(6,16)(7,15)(8,12)(11,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,6,7,3)(2,8,9,11)(4,12,10,14)(5,13,15,16)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=g,e^2=gh,f^2=gi,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, 8655102);