#8467009

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,e^2=ghi,f^2=gi,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, 8374205);