#501462

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 524 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=e,e^2=i,f^2=h,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, 408658);