#498051

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,22,28,34,40,30,17,20,29,32,18,23,25,19)(15,21,24,35,37,44,41,42,26,33,38,43,27,36,39,31);(5,7,9,8)(6,10,11,12)(14,19)(15,24)(16,25)(18,28)(20,30)(22,23)(26,38)(27,41)(29,40)(31,35)(32,34)(36,44)(37,39)(42,43);(5,8,9,7)(6,10,11,12)(14,20)(19,30)(21,33)(22,32)(23,34)(31,42)(35,43)(36,44);(13,15,25,39,18,27,29,38,17,26,40,41,28,37,16,24)(14,21,19,31,23,36,32,43,20,33,30,42,34,44,22,35)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405247);