#498313

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33579	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

626 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2179
maximal normal subgroup of	D₄×#2179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405509);