#428867

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

981 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:

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

Polycyclic presentation:

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

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, 336063);