#512300

order	512
nilpotency class	4
isoclinism	isoclinic class of #512070
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	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,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, 419496);