#8504771

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

3746 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)(4,9,12,14)(11,16,13,15)(17,18)(19,25)(20,30)(21,28)(22,26)(23,24)(27,31)(29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(4,12)(9,14)(11,13)(15,16)(18,24)(21,30)(25,32)(26,27)(34,38)(39,41)(40,45)(46,47);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,19,23,29)(18,25,24,32)(20,22,28,31)(21,27,30,26)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,20)(18,21)(19,22)(23,28)(24,30)(25,27)(26,32)(29,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,7,3)(2,10,6,8)(4,13,12,11)(9,16,14,15)(17,21)(18,20)(19,27)(22,25)(23,30)(24,28)(26,29)(31,32);(1,6,7,2)(3,10,5,8)(4,14,12,9)(11,16,13,15)(17,22)(18,26)(19,28)(20,29)(21,25)(23,31)(24,27)(30,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411967);