#8445966

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	#52765	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353162);