#8445869

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,9,4,6)(5,13,15,10)(11,14,12,16)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(34,35)(36,43)(40,42)(44,45);(2,8)(3,4)(10,13)(11,12)(19,29)(20,30)(24,28)(25,32)(33,34,38,35)(36,41,43,39)(37,40,46,42)(44,48,45,47);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36,38,43)(34,39,35,41)(37,45,46,44)(40,48,42,47);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,37,38,46)(34,40,35,42)(36,44,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=i,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, 8353065);