#510490

order	512
nilpotency class	4
isoclinism	isoclinic class of #510490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31665
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13334	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,15,21,31)(4,22)(5,10)(6,13)(7,12)(8,14)(9,18,29,19)(11,25)(16,28,26,17)(20,32,30,23)(24,27);(1,3,5,18,8,21,7,19)(2,9,12,31,14,29,10,15)(4,23,25,17,13,32,24,28)(6,26,11,30,22,16,27,20);(2,10)(3,16,21,26)(5,7)(6,27)(9,28,29,17)(11,22)(12,14)(15,23,31,32)(18,30,19,20)(24,25);(1,4)(2,11)(3,17,21,28)(5,24)(6,12)(7,25)(8,13)(9,26,29,16)(10,22)(14,27)(15,30,31,20)(18,23,19,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,c^2=eh,d^2=eh,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417686);