#6573662

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#453
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4647 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#3018, #72312788, #72313659
maximal normal subgroup of	D₄×#3018, #72312788, #72313659

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480858);