#430925

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1109 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338121);