#498015

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,d^c=di,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405211);