#501331

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

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, 408527);