#458071

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 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)(5,6,10,14)(7,12,20,31)(8,21,23,27)(9,13,26,24)(11,25,16,15)(17,33,35,18)(19,32,28,30)(22,36,34,29);(5,7,16,19,10,20,11,28)(6,12,25,30,14,31,15,32)(8,22,13,17,23,34,24,35)(9,18,27,29,26,33,21,36);(5,8,16,13,10,23,11,24)(6,9,25,27,14,26,15,21)(7,17,19,34,20,35,28,22)(12,29,30,33,31,36,32,18);(2,4)(5,9)(6,13)(7,18)(8,15)(10,26)(11,21)(12,17)(14,24)(16,27)(19,29)(20,33)(22,32)(23,25)(28,36)(30,34)(31,35)

Polycyclic presentation:

2^9|a^2=i,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, 365267);