#6381900

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60415	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2955, #72286365, #72286392
maximal normal subgroup of	D₄×#2955, #72286365, #72286392
maximal quotient of	#72286365, #72321848, #72322213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289096);