#503158

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61038	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410354);