#501059

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60444	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

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

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, 408255);