#501529

order	512
nilpotency class	4
isoclinism	isoclinic class of #501527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

601 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408725);