#512303

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|b^2=efi,d^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi

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, 419499);