#498548

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3147	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319298, #72319557, D₄×#2171
maximal normal subgroup of	#72319298, #72319557, D₄×#2171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405744);