#458565

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,16)(8,20,22,30)(9,24)(10,14)(11,15)(12,18)(13,23,29,21)(17,32,34,26)(19,27)(25,28)(31,33,36,35);(1,2)(3,4)(5,7,9,18,11,19,10,25)(6,12,14,16,15,28,24,27)(8,17,23,35,22,34,21,33)(13,26,30,31,29,32,20,36);(2,4)(8,21,22,23)(13,20,29,30)(17,33,34,35)(26,36,32,31);(5,8)(6,13)(7,17)(9,23)(10,21)(11,22)(12,26)(14,30)(15,29)(16,31)(18,35)(19,34)(20,24)(25,33)(27,36)(28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,d^c=de,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365761);