#458024

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

959 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:

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

Polycyclic presentation:

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

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, 365220);