#459763

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

926 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:

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

Polycyclic presentation:

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

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, 366959);