#501592

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

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

511 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,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, 408788);