#498344

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	#33600	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

621 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)(3,4)(5,6,10,15,11,8,22,24,12,16,28,34,27,25,9,14)(7,17,19,32,20,33,18,23,21,35,26,13,29,30,36,31);(5,7,11,29,12,21,27,20)(6,13,8,23,16,32,25,31)(9,26,10,18,22,19,28,36)(14,33,15,17,24,30,34,35);(2,4);(5,8,10,24,11,16,22,34,12,25,28,14,27,6,9,15)(7,17,26,13,20,33,36,31,21,35,19,32,29,30,18,23)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2179, #72319593, #72319601, …
maximal normal subgroup of	D₄×#2179, #72319593, #72319601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405540);