#8445885

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=gi,f^2=g,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, 8353081);