#498050

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, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,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, 405246);