#402284

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1007 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309480);