#512916

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319425, #72319605, D₄×#2263
maximal normal subgroup of	#72319425, #72319605, D₄×#2263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420112);