#505689

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319173, #72319567, #72319645
maximal normal subgroup of	#72319173, #72319567, #72319645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412885);