#499917

order	512
nilpotency class	4
isoclinism	isoclinic class of #499916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1440 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 180 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:

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

Polycyclic presentation:

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

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, 407113);