#8436652

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

3952 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=gh,e^2=g,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, 8343848);