#498286

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	#3035	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

625 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)(5,6,10,16,11,15,29,34,12,17,30,23,9,26,28,35)(7,13,19,31,20,27,33,36,21,8,22,25,18,24,14,32);(2,3)(5,7,10,19,11,20,29,33,12,21,30,22,9,18,28,14)(6,13,16,31,15,27,34,36,17,8,23,25,26,24,35,32);(5,8)(6,14)(7,16)(9,27)(10,25)(11,24)(12,13)(15,19)(17,33)(18,35)(20,34)(21,23)(22,26)(28,36)(29,32)(30,31);(1,3,4,2)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2171
maximal normal subgroup of	D₄×#2171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405482);