#458070

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

958 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)(6,12)(7,17)(9,10)(13,27)(14,26)(18,25)(19,32)(20,23)(21,24)(28,36)(29,33)(34,35);(5,6,9,16,11,12,10,15)(7,18,17,13,20,27,23,25)(8,14,21,31,22,26,24,30)(19,33,32,28,35,36,34,29);(5,7,9,17,11,20,10,23)(6,13,16,27,12,25,15,18)(8,19,21,32,22,35,24,34)(14,29,31,33,26,28,30,36);(2,4)(5,8)(6,14)(7,19)(9,21)(10,24)(11,22)(12,26)(13,28)(15,30)(16,31)(17,32)(18,33)(20,35)(23,34)(25,29)(27,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,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, 365266);