#10196068

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	#43400	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34,38,43)(35,39,46,40)(36,41,42,37)(44,45,47,48);(2,7)(4,8)(9,13)(12,14)(17,19,23,20)(18,24,22,25)(21,27,29,28)(26,31,30,32)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(2,7)(3,10)(9,13)(11,16)(33,35,38,46)(34,40,43,39)(36,44,42,47)(37,45,41,48);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,20,23,19)(18,25,22,24)(21,27,29,28)(26,31,30,32)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21,23,29)(18,26,22,30)(19,27,20,28)(24,31,25,32)(33,36)(34,41)(35,44)(37,43)(38,42)(39,45)(40,48)(46,47);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,44,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103264);