#498363

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#466	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

680 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)(9,10,12,18,25,30,38,27,13,16,26,29,14,19,23,15)(11,20,21,34,36,33,39,28,22,17,31,32,24,35,37,40);(5,7,8,6)(10,15)(11,21)(12,23)(14,25)(16,27)(18,19)(22,31)(24,39)(26,38)(28,32)(29,30)(33,35)(34,40)(36,37);(2,4)(10,16)(11,22)(15,27)(18,29)(19,30)(21,31)(24,36)(37,39);(9,11,23,37,14,24,26,31,13,22,38,39,25,36,12,21)(10,17,15,28,19,33,29,34,16,20,27,40,30,35,18,32)

Polycyclic presentation:

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

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, 405559);