#421905

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329101);