#498349

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,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, 405545);