#497722

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

700 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)(7,9)(8,10)(11,12)(13,14)(15,24)(16,19)(17,20)(18,32)(21,33)(22,28)(23,30)(25,35)(26,38)(27,37)(29,40)(31,34)(36,41)(39,43)(42,44);(5,7,8,11)(6,9,10,12)(14,19)(15,25)(16,28)(18,30)(20,34)(21,24)(22,23)(26,42)(27,43)(29,33)(31,41)(32,36)(37,38)(39,40);(6,10)(9,12)(14,20)(19,34)(21,37)(22,36)(23,32)(24,38)(35,44)(39,40);(13,15,17,26)(14,21,20,37)(16,29,31,43)(18,33,30,27)(19,35,34,44)(22,39,36,40)(23,24,32,38)(25,41,42,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 404918);