#497723

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×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 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)(14,20)(16,28)(17,30)(18,32)(21,34)(22,29)(23,35)(24,38)(25,40)(26,42)(31,33)(36,39)(37,44)(41,43);(5,7,8,11)(6,9,10,12)(13,14,19,22)(15,21,27,36)(16,29,30,20)(17,31,28,35)(18,33,32,23)(24,39,40,34)(25,41,38,44)(26,43,42,37);(6,10)(9,12)(13,15,19,27)(14,21,22,36)(16,24,30,40)(17,25,28,38)(18,26,32,42)(20,34,29,39)(23,37,33,43)(31,41,35,44);(13,16)(14,22)(15,24)(17,32)(18,28)(19,30)(20,35)(21,36)(25,42)(26,38)(27,40)(29,31)(34,44)(39,41)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^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, 404919);