#8437776

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3940 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=h,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, 8344972);