#512939

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
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 ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319455, #72319601, D₄×#2263, …
maximal normal subgroup of	#72319455, #72319601, D₄×#2263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420135);