#8463851

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,e^2=ghi,f^2=hi,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, 8371047);