#422304

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329500);