#481333

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=eg,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388529);