#6442604

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4667 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

(2,4)(5,6)(7,17)(8,13)(9,14)(10,15)(11,16)(12,18)(19,20);(5,7)(6,12)(8,16)(9,13)(10,18)(11,19)(14,17)(15,20);(1,2,3,4)(5,8)(6,13)(7,15)(9,20)(10,17)(11,12)(14,19)(16,18);(5,8)(6,13)(7,15)(9,20)(10,17)(11,12)(14,19)(16,18);(5,9,11,10)(6,14,16,15)(7,13,19,18)(8,20,12,17)

Polycyclic presentation:

2^9|c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#3018, #72312771, #72313632
maximal normal subgroup of	D₄×#3018, #72312771, #72313632

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349800);